Nonlinear Resonance Analysis: Theory, Computation, Applications

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N O N L I N E A R R E S O NA N C E A NA LY S I S Theory, Computation, Applications

Nonlinear resonance analysis is a unique mathematical tool that can be used to study resonances in relation to, but independently of, any single area of application. This is the first book to present the theory of nonlinear resonances as a new scientific field, with its own theory, computational methods, applications, and open questions. The book includes several worked examples, mostly taken from fluid dynamics, to explain the concepts discussed. Each chapter demonstrates how nonlinear resonance analysis can be applied to real systems, including large-scale phenomena in the Earth’s atmosphere and novel wave turbulent regimes, and explains a range of laboratory experiments. The book also contains a detailed description of the latest computer software in the field. It is suitable for graduate students and researchers in nonlinear science and wave turbulence, along with fluid mechanics and number theory. Color versions of a selection of the figures are available at www.cambridge.org/9780521763608. E LENA K ARTASHOVA is a Professor at the Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz. Her research interests include number theory, integrable dynamical systems, mathematical physics and their application in various areas, from fluid mechanics and meteorology to engineering. She was awarded the V. I. Vernadsky medal by the Russian Academy of Natural Sciences in 2009.

N O N L I N E A R R E S O NA N C E A NA LY S I S Theory, Computation, Applications E L E NA K A RTA S H OVA RISC, J. Kepler University, Linz, Austria

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521763608 © E. Kartashova 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN 13

978 0 511 90826 2

eBook (EBL)

ISBN 13

978 0 521 76360 8

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

In memory of my father, Lozanovsky Alexander Leonidovich

Contents

Preface Glossary

page ix xiv

1 Exposition 1.1 An easy start 1.2 Two classifications of PDEs 1.3 Hamiltonian formalism

1 1 11 19

2 Kinematics: Wavenumbers 2.1 An easy start 2.2 Irrational dispersion function, analytical results 2.3 q-class decomposition 2.4 Rational dispersion function 2.5 General form of dispersion function

30 30 32 45 57 60

3 Kinematics: Resonance clusters 3.1 An easy start 3.2 Topological structure vs dynamical system 3.3 Three-wave resonances 3.4 Four-wave resonances 3.5 NR-diagrams 3.6 What is beyond kinematics?

64 64 66 69 76 82 88

4 Dynamics 4.1 An easy start 4.2 Decay instability 4.3 A triad 4.4 Clusters of triads

90 90 93 96 106 vii

viii

4.5 4.6 4.7 4.8

Contents

A quartet Explosive instability NR-reduced numerical models What is beyond dynamics?

116 117 122 127

5 Mechanical playthings 5.1 Linear pendulum 5.2 Elastic pendulum

130 130 138

6 Wave turbulent regimes 6.1 An easy start 6.2 Quasi-resonances vs approximate interactions 6.3 Model of laminated turbulence 6.4 Energy cascades: dynamic vs kinetic 6.5 Rotational capillary waves 6.6 Discrete regimes in various wave systems 6.7 Open problems

144 144 147 150 158 161 170 176

7 Epilogue

182

Appendix Software References Index

185 209 221

Preface

Description of the universe in the scientific paradigm is based on conceptions of action and reaction. The main question then is: What sort of reaction should be expected to this or that action? Qualitatively, it looks logical to expect a bigger reaction to a bigger action, and this is mostly the case. But nature is not to be put into the Procrustes bed of our logical schemes, and a remarkable exception exists – the phenomenon of resonance. Resonance was first described by Galileo Galilei in 1638: “one can confer motion upon even a heavy pendulum which is at rest by simply blowing against it; by repeating these blasts with a frequency which is the same as that of the pendulum one can impart considerable motion” [73]. Nowadays resonance is generally regarded as a red thread that runs through almost every branch of physics; without resonance we would not have radio, television, music, etc. Resonance causes an object to oscillate; sometimes the oscillation is easy to see (vibration of a guitar string), but sometimes this is impossible without measuring instruments (electrons in an electrical circuit). Soldiers are commanded to break step while marching over a bridge, otherwise the bridge may collapse. Probably the most well-documented example of the resonance of a bridge is given by Tacoma Narrows Bridge, which was the third longest suspension bridge in the world in 1940. On the morning of November 7, 1940, the four-month old Tacoma Narrows Bridge began to oscillate dangerously up and down, tore itself apart, and collapsed over a period of about one hour. Though designed for winds of 120 mph, a wind of only 42 mph destroyed it. Experts agreed that somehow the wind caused the bridge to resonate and, nowadays, wind tunnel testing of bridge designs is mandatory. The very fortuitous fact for the history of science is that one professor of engineering decided to have a walk along the beach at the time when the oscillations began. He ran home, took a camera, and began to take a picture every five seconds. The pictures survived and have been turned into a video movie [227] showing the last few minutes before the catastrophe. ix

x

Preface

Other famous examples are the experiments of Tesla who studied experimentally in 1898 the vibrations of an iron column and noticed that at certain frequencies specific pieces of equipment in the room would start to jiggle. Playing with the frequency, he was able to move the jiggle to another part of the room. Completely fascinated with these findings, he forgot that the column ran downward into the foundation of the building, and the vibrations were being transmitted all over Manhattan. The experiments started a sort of a small earthquake in his neighborhood with smashed windows, swaying buildings, and panicky people in the streets. For Tesla, the first hint of trouble came when the walls and floor began to heave [36]. He stopped the experiment when he saw police rushing through the door. Generically, two types of resonance have to be distinguished: linear and nonlinear. From the physical point of view, they are defined by whether or not the external force coincides with the eigenfrequency of the system or not (linear and nonlinear resonance respectively). The condition of nonlinear resonance reads ωn = ω1 + ω2 + · · · + ωn−1 ,

(0.1)

with possibly different ωi = ω(ki ) being the eigenfrequencies of the linear part of some nonlinear partial differential equation. As will be explained in Chapter 1, the mathematical definition of resonance given above does not coincide with the physical one, which also includes resonance conditions on the wavevectors kn = k1 + k2 + · · · + kn−1 .

(0.2)

Wavevectors have integer coordinates in resonators which are bounded or periodical domains, while unbounded domains lead to real-valued wavevectors. Notice that “bounded or periodical” domain does not mean “small,” but rather indicates the importance of correlation between the domain size and the wavelengths. For instance, planetary waves in the Earth’s atmosphere (wavelengths ∼1000 km, [122]), Stokes edge waves in the coastal zone (wavelengths ∼10 m, [63]), and capillary waves in a cylindrical container with radius 200 mm (wavelengths ∼1 mm, [205]) all have integer wavevectors. Conditions of nonlinear resonance (0.1),(0.2), being regarded in integers, usually yield to solving Diophantine equations with many variables with huge degrees. This is equivalent to Hilbert’s 10th problem [89], which is proven to be algorithmically unsolvable [165]. Therefore, it is only in the last decade that nonlinear resonances have been studied independently in each application area. Analysis of nonlinear resonances presented in this book can be applied directly to the resonators of an arbitrary physical nature.

Preface

xi

Nonlinear resonances are ubiquitous in physics. Euler equations, with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.), all possessing nonlinear resonances [260]. Nonlinear resonances appear in numerous typical mechanical systems, such as an infinite straight bar, a circular ring, and a flat plate [136]. The so-called “nonlinear resonance jump,” important for the analysis of the turbine governor positioning system of hydroelectric power plants, can cause severe damage to mechanical, hydraulic, and electrical systems [91]. One tragic example is the collapse of the Sayano–Shushenskaya hydroelectric power station, Russia, on August 17, 2009, which cost not only enormous material losses but also 75 human lives. Nonlinear resonance is the dominant mechanism behind outer ionization and energy absorption in near-infrared laser-driven rare-gas or metal clusters [143]. Characteristic resonant frequencies observed in accretion disks allow astronomers to determine whether the object is a black hole, a neutron star, or a quark star [131]. Thermally induced variations of helium dielectric permittivity in superconductors are due to microwave nonlinear resonances [129]. Temporal processing in the central auditory nervous system analyzes sounds using networks of nonlinear neural resonators [5]. Nonlinear resonant response of biological tissue to the action of an electromagnetic field is used to investigate cases of suspected disease, e.g. cancer [234], etc. While linear resonances in various physical systems are presently well studied [198, 199], it is quite a nontrivial problem to compute the characteristics of nonlinear resonances or just to predict their very appearance, even in the one-dimensional case. Thus, the notorious Fermi-Pasta-Ulam numerical experiments [246] with a nonlinear one-dimensional string (carried out more than 50 years ago) are still not fully understood [16]. In these experiments, Fermi, Pasta, and Ulam simulated the vibrating string with quadratic and cubic nonlinearity by solving the system of nearest-neighbor coupled oscillators (32 and 64 oscillators in different series of experiments). Fermi thought, after many iterations, that the system would exhibit thermalization, i.e. a state of equipartition of energy, and would “forget” about initially exited oscillators. Instead, the system exhibited a puzzling quasi-periodic behavior. Keeping in mind the collapse of the Tahoma bridge, we can immediately see two main questions about nonlinear resonances we would like to have answers to: Where? and When? The answer to the first question is defined by the geometry of the physical system studied and is formulated mathematically in algebraic equations to be solved in integers. This part of the theory of nonlinear resonances is called kinematics. The answer to the second question is defined by the solutions to some systems of nonlinear ordinary differential equations; this part of the theory is called dynamics.

xii

Preface

This book is the first attempt to present the theory of nonlinear resonances, both kinematic and dynamic, as a new scientific area, with its own computational methods, applications, and open questions. It is written for an interdisciplinary audience and is structured as follows. Each chapter begins as simply as possible with an elementary section presenting the general ideas of the complete chapter. Thus, to get a notion about the general ideas and results presented in this book, read only the first sections of each chapter. Basic knowledge of linear algebra, differential equations, and a bit of common sense would be enough for understanding. Deeper reading demands additionally some knowledge of Hamiltonian formalism, number theory, graph theory, and theory of integrable systems. In Chapter 5, we show how to plan simple laboratory experiments with pendulums for observing physical manifestations of the mathematical notions and constructions introduced to describe nonlinear resonances. In Chapter 6, nonlinear resonance analysis is used for identifying and describing novel regimes in wave turbulent systems; important open problems are formulated at the end. The material presented in this book has been used since 2006 in a one-semester advanced course for undergraduates in pure and applied mathematics, and in computer science at J. Kepler University in Linz, Austria. I am deeply grateful to Vladimir Zakharov who encouraged my work on the discrete effects in wave turbulent systems. This work gave me the inspiration necessary to realize that nonlinear resonance analysis is a useful mathematical tool, as basic as linear Fourier analysis, but having as the application area weakly nonlinear partial differential equations. Very useful remarks, improvements and comments were received from Adrian Constantin, Jim Cooper, Vladimir Gerdjikov, Roger Grimshaw, Diane Henderson, Alexey Kartashov, German Kolmakov, Peter Lynch, Victor L’vov, Guenther Mayrhofer, Yuri Manin, Sergey Nazarenko, Mikhail Sokolovskiy, Efim Pelinovsky, Itamar Procaccia, Clemens Raab, Oleksii Rudenko, Veronika Retchitskaja, Jan Sanders, Michael Shats, Alexey Slunyaev, Lennart Stenflo, Vasilij Sotke, Igor Shugan, Rudolf Treumann, Mark Vilensky, and Erik Wahlén. I cordially thank them all. I am particularly indebted to Wolfgang Schreiner for developing a Web service [216] for nonlinear resonance computations and writing the Appendix “Software” where corresponding computer programs are presented and also the web-based service interface is described providing on-line access to these programs. The extracts included at the opening of Chapters 1–7 are reproduced, with permission, from M. Bulgakov, The Master and Margarita, translated by Richard Pevear and Larissa Volokhonsky (London: Penguin Classics, 2007), pp. 19, 28, 159, 179, 198, and 545. I owe very much to my son Peter who suffered a lot from the lack of my attention during the accomplishing of this text – suffered but never complained.

Preface

xiii

Finally, it was a pleasure and a privilege to work in close collaboration with Simon Capelin, Laura Clark, Megan Waddington, and Sehar Tahir at Cambridge University Press. All shortcomings of this book are my responsibility, of course.

Glossary

wavevector

k = (m, n), with m, n being integers as indexes of Fourier harmonics

dispersion function

ω = ω(k), ωj = ω(kj )

three-wave resonance conditions

ω1 + ω2 = ω3 , k1 + k2 = k3 , (*)

four-wave resonance conditions

ω1 + ω2 = ω3 + ω4 , k1 + k2 = k3 + k4 , (**)

exact resonance

solution of (*) or (**)

resonance solution set

all solutions of (*) or (**)

a triad

an exact solution of (*)

a quartet

an exact solution of (**)

resonance cluster, primary

a triad in three-wave system, a quartet in four-wave system

resonance cluster, generic

a set of primary clusters connected via common wavevector(s)

size of a cluster

number of connected primary clusters within generic cluster

geometrical structure (GS)

each k is shown as a node of integer lattice (m, n); nodes corresponding to one solution are connected by lines

topological structure

all topologically equivalent elements of GS are shown as one subgraph (resonance cluster) with number of xiv

Glossary

xv

appearances for each subgraph shown on the side NR-diagram

Graphical representation of a cluster in k-space; allows us to reconstruct uniquely its dynamical system

“slow” modes’ amplitudes physical variables Aj , j = 1, 2, 3, canonical variables Bj , j = 1, 2, 3, amplitude-phase Cj exp(iθj ), j = 1, 2, 3, dynamical system, primary three-wave case four-wave case

i B˙ 1 i B˙ 3 i B˙ 1 i B˙ 3

= ZB2∗ B3 , i B˙ 2 = ZB1∗ B3 , = −ZB1 B2 . = ZB2∗ B3 B4 , i B˙ 2 = ZB1∗ B3 B4 , = −ZB4∗ B1 B2 , i B˙ 4 = −ZB3∗ B1 B2 .

coupling coefficient

Z

dynamical phase of a triad of a quartet

ϕ12|3 = θ1 + θ2 − θ3 ϕ12|34 = θ1 + θ2 − θ3 − θ4

dynamical system, generic

few connected primary systems, corresponds to generic resonance cluster

resonance types

scale-resonances (three- and four-wave systems), angle-resonances (four-wave systems)

modes within a triad

A-mode, has maximal frequency ω3 , P-modes, have frequencies ω1 and ω2

mode pairs within a quartet

one-pairs: modes from one side of (**), (ω1 ,ω2 ) and (ω3 ,ω4 ) two-pairs: modes from different sides of (**); (ω1 ,ω3 ), (ω2 ,ω4 ), (ω1 ,ω4 ), (ω2 ,ω3 ),

connection types cluster of triads cluster of quartets

cluster’s reduction three-wave cluster four-wave cluster

AA-, AP- and PP-connections V-connection (via one mode), E-connection (via one-pair), D-connection (via two-pair) diminishing of the size of generic cluster due to the criterion of decay instability PP-reduction V-reduction

1 Exposition

“But there’s need for some proof. . .” Berlioz began. “There’s no need for any proofs,” replied the professor and he began to speak softly, while his accent for some reason disappeared: “It’s all very simple. . . M. Bulgakov Master and Margarita

1.1 An easy start As with every new theory, nonlinear resonance analysis has not come ex caelo. On the contrary, as Newton said: “If I have seen further it is only by standing on the shoulders of giants.” The giants to be grateful to now are Galileo Galilei (1564–1642), Jean Baptiste Joseph Fourier (1768–1830), and Jules Henri Poincaré (1854–1912). The father of modern physics, an Italian, Galileo Galilei, fascinated by the movement of a simple pendulum, identified one of the most important natural phenomena – resonance. French politician and mathematician Fourier, trying to understand what happens when a hot piece of metal rod is put into water, developed the mathematical apparatus – Fourier analysis – for describing solutions to linear partial differential equations (PDEs), without which no area of contemporary science is conceivable. Another French mathematician, Henri Poincaré, used Fourier analysis in order to give a strict mathematical definition of resonance and developed a method – Poincaré transformation – allowing, under some assumptions, to reduce the search for solutions to a nonlinear PDE to the search for resonances. This way the foundations of nonlinear resonance analysis were laid. Below we begin with the mathematical problem and will return to the pendulum in Chapter 5, for this is a very handy object to illustrate even the quite complicated mathematical results presented in this book.

1

2

Exposition

Fourier analysis Thus, in the beginning was Fourier analysis. More precisely, in the beginning was a discussion. From 1750–1760, d’Alembert, Euler, Bernoulli, Clairaut, and Lagrange were involved in a prolonged and heated controversy about solutions to the equation of a vibrating string: ψtt − α 2 ψxx = 0

(1.1)

with some constant α. D’Lambert and Euler derived a functional form of the solution (1.2) ψ = ϕ1 (x + αt) + ϕ2 (x − αt) (on an infinite line), while Bernoulli was the first to present the solution in the form of a trigonometric series of sines and cosines of multiple variables: ψ = A1 sin x cos αt + A2 sin 2x cos 2αt + · · ·

(1.3)

Bernoulli stated that his solution (1.3) included solution (1.2) as a particular case. Euler disagreed testily, his arguments being that if this were true, an arbitrary function of one variable could be presented as a series of sines, which obviously is not possible because an arbitrary function is not necessarily odd and periodic. With the discussion at this stage, the young and still unknown mathematician Lagrange appeared on the scene and tried to prove that the solution in the functional form (1.2) is more general than Bernoulli’s trigonometric series (1.3). By an irony of fate, Lagrange did not notice that at some intermediate step in his computations he actually derived the explicit form of the coefficients for Bernoulli’s presentation, a slightly different version from the form usual nowadays. Crucial progress in this discussion was achieved half a century later, due to the French mathematician, Joseph Fourier. This same Joseph Fourier, who had already taken part in the promotion of the French revolution, was the governor of Lower Egypt and secretary of the Cairo Institute, had a Chair in Mathematics in L’Ecole Polytechnique, etc. In 1811, Fourier submitted his paper on the theory of heat conduction (Mémoire sur la propagation de la chaleur) to the Paris Academy, as a candidate for the Great Prize in Mathematics (Grand Prix de Mathématiques) for the year 1812. Fourier derived the heat equation ψt − αψxx = 0,

(1.4)

developed the method of separation of variables to solve it, and laid the foundations for what is now known as Fourier analysis. The new-born baby had its problems: although Fourier got the prize, it was accompanied by a lot of criticisms from the

1.1 An easy start

3

referees. The list of referees included Lagrange, Laplace, and Legendre. The list of criticisms included accusations of the absence of rigor in his analysis, stating that “the manner in which the author arrives at these equations is not exempt from difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor.” A very bitter pill, accompanying the award, was the fact that Fourier’s paper was not published in the Proceedings of the Academy (Mémoires de l’Académie des Sciences) till 1822, when Fourier was already the Secretary of the Academy. In all fairness, we should notice that the publication was arranged by Delambre, not by Fourier himself. But the publication did not improve the situation much, and for the last eight years of his life he still had to tackle the assaults of Biot and Poisson; the only difference from the past was that the direction of the attacks changed. In 1810, the main target of Fourier’s opponents was the incorrectness of his results, while 20 years later it suddenly became the matter of priority. Another half a century and the efforts of Cauchy, Dirichlet, Riemann, and many other great mathematicians were needed to figure out the conditions of convergency for Fourier integrals under different assumptions, necessary and sufficient conditions for various classes of (piecewise) continuous functions to have Fourier presentation, etc. Armed with our present knowledge, we can say that for the simplest possible  2πexample – a periodic function ψ(x) of one variable, with period 2π and with 0 ψ 2 (x)dx < ∞ – two equivalent Fourier presentations can be given in trigonometrical and complex form: ∞

a0  + (an cos nx + bn sin nx), ψ(x) ∼ 2

(1.5)

n=1

or ψ(x) ∼

∞ 

cn exp (inx),

(1.6)

−∞

with coefficients an , bn , cn given below:  1 π a0 = ψ(x)dx, π −π  1 π ψ(x) cos nxdx, an = π −π  1 π ψ(x) sin nxdx, bn = π −π and c0 =

a0 , 2

cn =

an − ibn , 2

c−n =

an + ibn , 2

(1.7) (1.8) (1.9)

(1.10)

4

Exposition

with n = 1, 2, 3, . . . , which follows from the Euler presentation exp (ix) = cos x + i sin x,

exp (−ix) = cos x − i sin x.

(1.11)

Each of the series (1.5), (1.6) converges to ψ(x) in the mean; if ψ(x) is a continuously differentiable function, its Fourier series converges uniformly. For a function of multiple variables, similar formulas can be obtained. One of the most important developments of Fourier analysis, from the computational point of view, was the question of whether or not it is possible to approximate an arbitrary function by a finite trigonometrical Fourier sum. The answer given by Weierstrass in 1885 was positive: for any periodic continuous function ψ(x) defined on a compact and any arbitrary small number ε such that 0 < ε  1, there exists a finite Fourier sum SN , though not unique, N 0,

(1.18)

and the initial conditions for the temperature along the bar ψ(x, 0) = ϕ(x),

t = 0, 0 < x < l.

(1.19)

Assume that the solution to (1.4) has the form ψ(x, t) = X(x)T (t)

(1.20)

and substitute (1.20) into (1.4) XTt = αXxx T ⇒

Xxx Tt = ≡λ αT X

(1.21)

and obviously λ ≡ const. This way the PDE is reduced to two ordinary differential equations (ODEs) Tx = αλT , and Xxx = λX. (1.22) Let λ be real, and regard cases: Case 1: λ > 0. Solutions have the form T (t) = A exp (αλt) and

√

√

X(x) = B exp x λ − C exp − x λ .

(1.23)

(1.24)

6

Exposition

The boundary conditions (1.18) are satisfied if and only if X(0) = X(l) = 0, i.e. √

√

0 = X(0) = B + C, 0 = X(l) = B exp l λ + C exp − l λ , √

√

√

⇒ exp l λ + exp − l λ = 0 ⇒ sinh l λ = 0, (1.25) which is not possible. Case 2: λ = 0. This yields Tt = 0,

Xxx = 0 ⇒ T (t) = A,

X(x) = B + Cx,

(1.26)

and imposing the boundary conditions gives 0 = X(0) = B,

0 = X(l) = Cl ⇒ B = C = 0,

(1.27)

i.e. also in this case there are no nontrivial solutions. Case 3: λ < 0. Let λ = −μ2 with a real μ, then again the boundary conditions (1.18) yield exp (iμl) + exp (−iμl) = 0,

i.e.

sin (μl) = 0.

(1.28)

The solutions of (1.28) have the form μl = nπ,

n = ±1, ±2, . . .

(1.29)

and correspondingly 

αn2 π 2 t , Tn (t) = exp − l2  

nπ x  inπ x inπ x , − exp − = 2i sin Xn (x) = exp l l l

(1.30) (1.31)

with n = 1, 2, . . . . By superposition, the final form of the solution satisfying the boundary conditions reads ψ(x, t) =

∞  n=1

 αn2 π 2 t nπ x , bn exp − sin l2 l

(1.32)

while the initial conditions (1.19) must have a form ψ(x, 0) = ϕ(x) =

∞  n=1

bn sin

nπ x , l

for

0 < x < l.

(1.33)

1.1 An easy start

7

By imposing different boundary conditions, namely ψx (0, t) = ψx (l, t) = 0,

t > 0,

(1.34)

instead of (1.18), another solution can be deduced ψ(x, t) =

∞  n=1



αn2 π 2 t an exp − l2

cos

nπ x , l

(1.35)

0 < x < l.

(1.36)

with initial conditions given by ψ(x, 0) = ϕ(x) =

∞  n=1

an cos

nπ x , l

for

Resonance conditions As we have seen above, the linearity of a differential operator is important when applying Fourier analysis. On the other hand, a great majority of physically relevant differential equations are nonlinear. In order to show that nonlinear PDEs in physics are not exotic, we follow [247] describing a class of problems that leads immediately to some nonlinear PDE. In many physical problems, we need to find some relation between two quantities, say unit density ρ of some physical entity and its unit flow ψ, so that the velocity of a flow could be defined as ψ/ρ. Both ρ and ψ are functions of space and time variables, ρ = ρ(x, t) and ψ = ψ(x, t). Most physical systems have some conservation law that could be written as  x2 d ρdx + ψ(x2 , t) − ψ(x1 , t) = 0 (1.37) dt x1 for any fixed interval x1 , x2 , as shown in Fig. 1.1. And if ρ and ψ are smooth, then in the limit x1 → x2 and the conservation law takes a form ρt + ψx = 0.

(1.38)

Very often there exist some intuitive or empirical considerations allowing us to regard flow as a function of density, i.e. ψ = F (ρ), which yields immediately a nonlinear PDE ρt + c(ψ)ψx = 0 (1.39) with c(ψ) = Fx (ρ). This PDE describes many various physical and technical problems, such as flood waves in rivers (wave height h plays the role of density and

8

Exposition

Fig. 1.1 Conservation law

c(h) is the flow velocity), transport flow (ρ now is the number of cars at the unit length of a highway, ψ is the number of cars passing a line x in unit time, and the highway has no outlets), erosion of mountains, chemical exchange processes, absorption, etc. The natural question to ask is when is it possible to linearize a given nonlinear differential equation by an appropriate invertible change of variables. It is easier to answer this question beginning with a nonlinear ODE example. Example 2: Elimination of the quadratic term Consider

ψ˙ = λψ + ψ 2

(1.40)

and try to transform it into the linear form ϕ˙ = λϕ

(1.41)

by the change of variables ψ = ϕ + aϕ 2 . Here a is an unknown parameter and should be chosen in such a way that the term ψ 2 in (1.40) disappears. It is easy to see that ψ˙ = ϕ˙ + 2aϕ ϕ˙ = (1 + 2aϕ)ϕ, ˙ ⇒ (1 + 2aϕ)ϕ˙ = λ(ϕ + aϕ 2 ) + (ϕ + aϕ 2 )2 = λ(1 + 2aϕ)ϕ + (1 − λa)ϕ 2 + p(ϕ) ⇒ 1 − λa 2 1 ϕ˙ = λϕ + ϕ + p(ϕ). 1 + 2aϕ 1 + 2aϕ

(1.42)

The use of a series presentation of (1 + 2aϕ)−1 over the powers of ϕ allows us to conclude that (1+2aϕ)−1 p(ϕ) contains only the terms with ϕ 3 , ϕ 4 , . . . . If λ = 0, then the choice a = 1/λ gives

1.1 An easy start

ϕ˙ = λϕ +

1 p(ϕ). 1 + 2aϕ

9

(1.43)

Similarly, the change of variables ϕ = ζ + ζ 3 will annihilate the term with ϕ 3 and so on. Moreover, a similar procedure can be applied for systems of nonlinear ODEs. Indeed, let us consider a system of nonlinear ODEs of the simple form  ψ˙ j = λj ψj + aj (m)ψ m , j = 1, 2, . . . , N (1.44) with multiple indexes, i.e. m ≡ (m1 , . . . , mN ), aj ≡ aj (m1 , . . . , mN ), ψ m ≡ ψ m1 · . . . ψ mN , and the minimal term in this sum is of the order ≥ 2, i.e. m1 + · · · + mN = |m| ≥ 2. Again, a simple change of variables  bj (m)ϕjm (1.45) ψj = ϕj + |m|≥2

with some (not yet known) coefficients bj , yields a new system  ϕ˙j = λj ϕj + cj (m)ϕjm .

(1.46)

|m|≥2

Let us try to choose the coefficients bj in such a way that cj (m) = 0 ∀j, m. If this is possible, the change of variables (1.46) will linearize the system of nonlinear ODEs (1.44). Direct substitution of (1.46) into (1.44) allows us to determine bi  only if the corresponding coefficient −λi + N j =1 (mj λj ) = 0. Otherwise, (1.46) remains nonlinear. Definition 1. If there exist N positive integers m1 , m2 , . . . , mN , such that N  j =1

mj ≥ 2

and

N 

mj λj − λi = 0,

(1.47)

j =1

then (1.47) are called in mathematics the resonance conditions for the system (1.44). The number N is called the order of resonance. Definition 2. The change of variables (1.45) is called the Poincaré transformation. Theorem 1 (The Poincaré theorem on linearization of the vector field). If the resonance conditions (1.47) are not fulfilled, then (1.44) can be linearized by an appropriate Poincaré transformation. Definition 3. If the resonance conditions (1.47) are fulfilled for some set of integers m1 , . . . , mN , then (1.46) is called the normal form of (1.44).

10

Exposition

The normal form (1.46) is equivalent to (1.44) with all nonresonant terms being eliminated by the Poincaré transformation. Usually, the general form of bj is not known and is computed order by order. The combination of the Poincaré theorem with the Poincaré transformation yields a fact of utmost importance, which is indeed sound ground for the entire nonlinear resonance analysis: A system of nonlinear ODEs, if not linearizable, can be transformed into normal form, with the resonance term having the smallest order. To see the significance of this issue, let us suppose that ϕ in (1.42) is small, 0 < ϕ ∼ ε  1, and the term with ϕ 2 ∼ ε2 is resonant. Then all other terms, being of order ε3 , ε4 , . . . , can be neglected and a solution to ϕ˙ = λϕ +

1 − λa 2 ϕ 1 + 2aϕ

(1.48)

will give an approximate solution to (1.46), with the terms of the next order of smallness omitted. Notice that the form of equations (1.40) and (1.48) differs only by the coefficient in front of the second-order term: ψ˙ = λψ + ψ 2

and ϕ˙ = λϕ +

1 − λa 2 ϕ , 1 + 2aϕ

(1.49)

with ψ = ϕ + O(ε3 ). Of course, most physically relevant equations are PDEs, not ODEs. However, the notion of resonance turned out to be so important that physical classifications of PDEs [248] have been developed based on the fact of whether or not a PDE might possess some resonance. This classification, different from standard mathematical classification, will be presented in the Section 1.2. The next step – to proceed from a nonlinear PDE to a system of nonlinear ODEs – can be performed by a number of variation and perturbation methods, an example is given in Section 1.2. Hamiltonian formalism is briefly sketched in Section 1.3 for it allows us to simplify and standardize all these methods and obtain the universal form of dynamical equations, independent of the details of the physical system. These equations are written using canonical variables, which can be transformed back to physical variables using Fourier transformation. This way Poincaré’s approach works perfectly also for the case of PDEs, modulo zero denominators in (1.48). The problem known nowadays as the small divisor problem was regarded by Poincaré as “the fundamental problem of dynamics” [155]. This book is devoted to the study of what is happening when divisors are zero or small enough to cause the problem while applying asymptotical methods.

1.2 Two classifications of PDEs

11

The main goal we pursue in this book is to collect the results on nonlinear resonances obtained over the last 20 years, scattered in the numerous, partly unavailable publications, and also differing sometimes in definitions and notations. We try to put the available knowledge into a general context, which will allow readers to regard nonlinear resonance analysis as a handy apparatus while investigating resonances in various application areas. We will learn how: • to find solutions to resonance conditions (Chapter 2); • to construct resonance clustering, represent it graphically as a set of NR-diagrams, and

to deduce from it the form of dynamical system for each cluster (Chapter 3); • to solve these dynamical systems (Chapter 4); • to plan simple laboratory experiments for observing the real-world occurrence of

mathematical entities introduced in previous chapters (Chapter 5); • to apply nonlinear resonance analysis for describing various phenomena discovered in

wave turbulent systems (Chapter 6); • to implement all these mathematical methods as an on-line computational service (Appendix).

1.2 Two classifications of PDEs Mathematical classification Classical mathematical classification of PDEs is based on a form of equations presented as follows. For a PDE of second order on two variables, aψxx + bψxy + cψyy = F (x, y, ψ, ψx , ψy );

(1.50)

its characteristic equation is written as dx b 1 2 = ± b − 4ac dy 2a 2a

(1.51)

and three types of PDEs are defined: • b 2 < 4ac, elliptic PDE:ψxx + ψyy = 0, • b 2 > 4ac, hyperbolic PDE: ψxx − ψyy = 0, • b 2 = 4ac, parabolic PDE: ψxx − ψ = 0.

Nonlinearity can be taken in an arbitrary form. A “bad” example of a Tricomi equation yψxx + ψyy = 0 (1.52)

12

Exposition

shows immediately the incompleteness of this classification even for second-order PDEs, because a PDE can change its type depending, for instance, on the initial conditions. Each type of PDE demands then special types of initial and boundary conditions for the problem to be well-posed. Mathematical classification can be generalized to PDEs of more variables but not to PDEs of higher order.

Physical classification Contrary to mathematical classification, physical classification of PDEs is based not on the form of equations, but on the form of solutions. In this case, a PDE is regarded in the very general form, without any restrictions on the number of variables or the order of equations. On the other hand, the necessary preliminary step in this classification is the partitioning of all the variables into two groups – time- and space-like variables. This division originates from special relativity theory, where time and three-dimensional space are treated together as a single four-dimensional Minkovski space. In Minkovski space, a metric allowing us to compute an interval s along a curve between two events is defined analogously to distance in Euclidean space: ds 2 = dx 2 + dy 2 + dz2 − c2 dt 2 ,

(1.53)

where c is the speed of light, and x, y, z, and t denote respectively space and time variables. Notice that although in mathematical classification all variables are treated equally, obviously the results can be used in any application only after similar partitioning of variables has been done. Let us now assume that a linear PDE in one space variable with constant coefficients has a wave-like solution, i.e. a solution in the form of the Fourier harmonic ψ(x, t) = A exp i[kx − ωt] (1.54) with amplitude A, wavenumber k, and wave frequency ω. Then the substitution of ∂t = −iω, ∂x = ik transforms the linear PDE into a polynomial on ω and k, for instance: ψt + ψx + ψxxx = 0 ⇒ ω(k) = k − k 3 . (1.55) Definition 4. A real-valued function ω = ω(k) such that d2 ω/dk 2 = 0 is called a dispersion function [248]. A linear PDE with a wave-like solution is called an evolutionary dispersive LPDE. A nonlinear PDE with a dispersive linear part is called an evolutionary dispersive NPDE.

1.2 Two classifications of PDEs

13

The word “evolutionary” reminds us that variables are already partitioned into space and and time variables, and both types are present. If there is no time variable in a PDE, it is called stationary. The notion of a dispersion function can easily be generalized for the case of many space variables, namely x1 , . . . , xh . Indeed, the linear part of the initial PDE then takes the form  ∂ ∂ ∂ , ,..., P (1.56) ∂t ∂x1 ∂xh and, correspondingly, the dispersion function can be computed from P (−iω, ik1 , . . . , ikh ) = 0

(1.57)

with polynomial P . In this case, we will have not a wavenumber k but a wavevector k = (k1 , . . . , kh ) and the condition of the nonzero second derivative of the dispersion function takes a matrix form: ⏐ 2 ⏐ ∂ ω ⏐ ⏐ ∂k ∂k i

j

⏐ ⏐ ⏐ = 0. ⏐

(1.58)

The given dispersion function allows us to reconstruct the corresponding linear PDE. Of course, this presentation is simplified in many respects; in particular, the existence of boundary conditions is not taken into account. Depending on the actual boundary conditions, normal modes of the linearized problem may differ from the propagating plane waves given by the Fourier harmonic (1.54). For instance, zero boundary conditions in a rectangular box typically, though not always, lead to standing waves, ψ(x, t) = |Ak | sin(k · x + θsp ) sin(ωt + θt ),

(1.59)

where θsp and θt are the space and time phases respectively. A more complicated form of the normal mode is given by atmospheric planetary motions with zero boundary conditions on a sphere (atmospheric planetary waves [219]):  APnm (sin ϕ) exp i

 2m t , mλ + n(n + 1)

(1.60)

with two spherical space variables, the latitude ϕ, −π/2 ≤ ϕ ≤ π/2, and the longitude λ, 0 ≤ λ ≤ 2π . Here A is the constant wave amplitude, ω = −2βm/ [n(n+1)], Pnm (x) is the associated Legendre function of degree n and order m, and β is the dimensional derivative of the Coriolis parameter with respect to latitude.

14

Exposition

Moreover, the same equation with zero boundary conditions in a rectangular domain [x, y] (ocean planetary waves) yields a different form of the eigenfunction [101, 128]:      mx ny β A sin π x + ωt (1.61) sin π exp i Lx Ly 2ω with ω = β/[π(m2 + n2 )]. The notion of a linear wave used further on in this book, though not strictly defined [248], can be regarded as a generic notion for both the Fourier harmonic and normal mode. General methods for computing normal modes of a linear problem with given boundary conditions can be found in numerous textbooks on the theory of differential equations. The explicit form of normal modes and corresponding dispersion functions can be found in the literature with numerous physically relevant examples (e.g. in [192] normal modes are constructed for ocean planetary waves in a rectangular and in a circular domain with zero boundary conditions). This way all PDEs are divided into two classes – dispersive and non-dispersive [248]; comprehensive mathematical treatment of the dispersive equations can be found in [67]. A few well-known examples of evolutionary dispersive nonlinear PDEs are given below, which cover various application areas from biology to fluid mechanics to plasma physics to medicine to astronomy: • Boussinesq equation:

ψt t − (ψxx + ψ 2 )xx = 0;

(1.62)

• Hasegawa–Mima equation:

(ψ)t + βψx + J (ψ, ψ) = 0;

(1.63)

• Kadomtsev–Petviashvili equation:

(ψt + 6ψψx + ψxxx )x + 3ψyy = 0;

(1.64)

• Korteweg–de Vries equation:

ψt + ψxxx − 6ψψx = 0;

(1.65)

iψt + ψxx + f (|ψ|)ψ = 0;

(1.66)

ψt = εψ − (2 + 1)2 ψ + g1 ψ 2 − ψ 3 ;

(1.67)

• Schrödinger equation: • Swift–Hohenberg equation:

• Zakharov system of equations:

iψt + ψxx − ψϕ = 0,

ϕt t − ϕxx − |ψ|2xx = 0.

(1.68)

1.2 Two classifications of PDEs

15

Now we can introduce the notion of nonlinear resonance in a physical way. Remembering Galileo Galilei, let us consider first a linear oscillator, or pendulum, driven by a small force (1.69) xtt + p 2 x = εeit . Here p is the eigenfrequency of the system,  is the frequency of the driving force, and 0 < ε  1 is a small parameter. Deviation of this system from equilibrium is small (of order ε), if there is no resonance between the frequency of the driving force εeit and the eigenfrequency of the system. If these frequencies coincide, then the amplitude of the oscillator grows linearly with time and this situation is called resonance in physics. Let us now regard an arbitrary (weakly) nonlinear PDE of the form L(ψ) = −εN (ψ),

(1.70)

where L is an arbitrary linear dispersive operator and N is an arbitrary nonlinear operator. Any two solutions of L(ψ) = 0 can be written out as A1 exp i[k1 x − ω(k1 )t]

and A2 exp i[k2 x − ω(k2 )t]

(1.71)

with constant amplitudes A1 , A2 . Intuitively, the natural expectation is that solutions of weakly nonlinear PDE will have the same form as linear waves, i.e. Fourier harmonics, but perhaps with amplitudes depending on time. Taking into account that nonlinearity is small, each amplitude is regarded as a slow-varying function of time, i.e. Aj = Aj (t ε). Standard notation is Aj = Aj (T ), where T = t ε is called slow time while time t in this context is called fast time. Unlike linear waves, for which their linear combination was also the solution to L(ψ) = 0 due to the superposition principle, it is not the case for nonlinear waves. Indeed, substitution of two linear waves into the operator εN (ψ) generates terms of the form exp i{(k 1 + k 2 )x − [ω(k 1 ) + ω(k 2 )]t}, (1.72) which play the role of a small driving force for the linear wave system similar to the case of the linear oscillator above. This driving force produces a small effect on the wave system till resonance occurs, i.e. till the wavenumber and the wave frequency of the driving force do not coincide with some wavenumber and some frequency of the eigenfunction: ω 1 + ω2 = ω3 , where notation ωi = ω(k i ) is used.

k1 + k2 = k3 ,

(1.73)

16

Exposition

Definition 5. Equations (1.73) are called in physics the resonance conditions for three-wave processes. Correspondingly, the resonance conditions for s-wave processes, s < ∞, read as ω1 ± ω2 ± · · · ± ωs = 0,

k 1 ± k 2 ± · · · ± k s = 0.

(1.74)

Notice that coordinates of wavevectors are integers, being indexes of Fourier harmonics. Multi-scale method Solutions to resonance conditions (1.73), if any, provide coordinates of wavevectors for three resonantly interacting waves A1 (T ) exp i[k 1 x − ω(k 1 )t],

(1.75)

A2 (T ) exp i[k 2 x − ω(k 2 )t],

(1.76)

A3 (T ) exp i[k 3 x − ω(k 3 )t],

(1.77)

and we can see immediately that one main goal – to find solutions of an evolutionary nonlinear dispersive PDE – has not yet been achieved. We have found all k i but functions A1 (T ), A2 (T ), and A3 (T ) also have to be found. This could be done, for example, by one of the perturbation methods, which are much in use in physics and when dealing with equations that have some small parameter 0 < ε  1 [178]. The main idea of such a method is very simple – an unknown solution, depending on ε, is written out in a form of infinite series on different powers of ε, and thus the coefficients in front of any power of ε are computed. Then an approximate solution ψ˜ to equation (1.70) is constructed as a combination of Fourier harmonics with amplitudes A, depending on the slow time T . Example 3: Multi-scale method for barotropic vorticity equation Let us demonstrate by taking as an example the barotropic vorticity equation (BVE) on a sphere ∂ψ ∂ψ +2 + εJ (ψ, ψ) = 0, ∂t ∂λ

(1.78)

where 1 ∂ 2ψ ∂ψ ∂ 2ψ , + − tan ϕ 2 2 2 ∂ϕ cos ϕ ∂λ ∂ϕ  ∂a ∂b 1 ∂a ∂b − J (a, b) = cosϕ ∂λ ∂ϕ ∂ϕ ∂λ

ψ =

(1.79) (1.80)

and the linear part of the spherical BVE has wave solutions in the form (1.60).

1.2 Two classifications of PDEs

17

First, let us search for a solution to the form ψ = ψ0 (λ, ϕ, t, T ) + εψ1 (λ, ϕ, t, T ) + ε 2 ψ2 (λ, ϕ, t, T ) + . . . ,

(1.81)

then zero approximation ψ0 is given as a sum of three linear waves ψ0 (λ, ϕ, t, T ) =

3 

Ak (T )P (k) exp(θk )

(1.82)

k=1

with notations P (k) = Pnmk k and θk = mk λ − ωk t. Then ⎧ 0 ⎪ ⎨ε : ε1 : ⎪ ⎩ 2 ε :

∂ψ0 /∂t + 2∂ψ0 /∂λ = 0, ∂ψ1 /∂t + 2∂ψ1 /∂λ = −J (ψ0 , ψ0 ) − ∂ψ0 /∂T ,

(1.83)

(. . .).

Notation (. . .) in the last line means that the corresponding terms are omitted. Now 3    ∂ψ0 =i P (k) mk Nk Ak exp(iθk ) − A∗k exp(−iθk ) ; ∂λ

(1.84)

3    d ∂ψ0 =− Nk P (k) cos φ Ak exp(iθk ) + A∗k exp(−iθk ) ; ∂ϕ dϕ

(1.85)

k=1

k=1

3 

J (ψ0 , ψ0 ) = −i

j,k=1

Nk mj P (j )

 d (k)  P Aj exp(iθj ) − A∗j exp(−iθj ) dϕ

3    d × Ak exp(iθk ) + A∗k exp(−iθk ) + i Nk mk P (k) P (j ) dϕ j,k=1    × Aj exp(iθj ) + A∗j exp(−iθj ) Ak exp(iθk ) − A∗k exp(−iθk ) ;

(1.86)   3 dA∗k ∂ψ0  (k) dAk = exp(iθk ) + exp(−iθk ) P Nk ∂T dT dT

(1.87)

k=1

(with notation Nk = nk (nk + 1)) yields the condition of unbounded growth of the left-hand side in the form J (ψ0 , ψ0 ) =

∂ψ0 ∂T

(1.88)

18

Exposition

with resonance conditions θj + θk = θi ∀ j, k, i = 1, 2, 3. Let us fix some specific resonance condition, say, θ1 + θ2 = θ3 , then   dA∗3 ∂ψ0 ∼ (3) dA3 exp(iθ3 ) + exp(−iθ3 ) , (1.89) = −N3 P ∂T dT dT  d (2) d (1) ∼ P − m1 P (1) P (2), (1.90) J (ψ0 , ψ0 ) = −i(N1 − N2 ) m2 P dϕ dϕ (1.91) A1 A2 exp[i(θ1 + θ2 )] − A∗1 A∗2 exp[−i(θ1 + θ2 )], where notation ∼ = means that only those terms are written out that can generate the chosen resonance. Let us substitute these expressions into the coefficient by ε 1 , i.e. into equation ∂ψ1 ∂ψ0 ∂ψ1 +2 = −J (ψ0 , ψ0 ) − , (1.92) ∂t ∂λ ∂T multiply both parts of it by   P (3) sin ϕ A3 exp(iθ3 ) + A∗3 exp(−iθ3 )

(1.93)

and integrate all over the sphere with t → ∞. As a result, the following two equations can be obtained: dA3 = 2iZ(N2 − N1 )A1 A2 , dT dA∗ N3 3 = −2iZ(N2 − N1 )A∗1 A∗2 , dT N3

(1.94) (1.95)

where  Z=

π/2

−π/2

 m2 P

(2)

 d (1) d (3) (1) d (2) P − m1 P P P dϕ. dϕ dϕ dϕ

(1.96)

The same procedure obviously provides similar equations for A2 and A3 , if we fix the corresponding resonance conditions: dA1 dT dA∗1 N1 dT dA2 N2 dT dA∗2 N2 dT N1

= −2iZ(N2 − N3 )A3 A∗2 ,

(1.97)

= 2iZ(N2 − N3 )A∗3 A2 ,

(1.98)

= −2iZ(N3 − N1 )A∗1 A3 ,

(1.99)

= 2iZ(N3 − N1 )A1 A∗3 .

(1.100)

1.3 Hamiltonian formalism

For simplicity, the resulting system is written in the form ⎫ ⎧ ˙ 1 = −2iZ(N2 − N3 )A3 A∗ ,⎪ A N ⎪ 1 2 ⎬ ⎨ N2 A˙ 2 = −2iZ(N3 − N1 )A∗1 A3 , + c.c. ⎪ ⎪ ⎭ ⎩ N3 A˙ 3 = 2iZ(N2 − N1 )A1 A2 ,

19

(1.101)

with “c.c.” meaning “complex conjugate.” Coefficients of (1.101) are called coupling coefficients and should be nonzero. Otherwise, four-wave resonances have to be regarded, i.e. the sum in (1.82) will have four terms, not three. The procedure described above is quite straightforward and can be programmed in any symbolic language (an example of its implementation is given in the Appendix). As we began with the PDE (1.78) in physical variables ϕ and λ, the coefficients of the dynamical system (1.101) depend on the explicit form of the corresponding eigenmodes (1.60), (1.61), etc. On the other hand, Hamiltonian formalism yields the general form of these systems for a large class of PDEs, and allows us in particular to study the dynamics of resonances, independent of the details of the physical system where they occur. 1.3 Hamiltonian formalism Notion of Hamiltonian Hamiltonian formalism is a reformulation of the Lagrangian dynamics in new variables, called nowadays canonical variables q and p (corresponding to the action-angle variables in classical mechanics) instead of classical space variables. Dynamical equations then read q˙ =

∂H , ∂p

p˙ = −

∂H ∂q

(1.102)

and can be regarded as a generalization of Newton’s second law F = ma =

dp dt

(1.103)

to systems where momentum p = m v, with m, a, and v being acceleration, mass, and velocity respectively. As a rule, a Hamiltonian can be interpreted as the total energy of the system. To demonstrate this, let us regard an example of a simple linear pendulum. Example 4: Linear pendulum Consider a linear pendulum of mass m and length l, fixed at one end and swinging under gravity force, assuming that all the mass is on the other end. Let θ be an angle

20

Exposition

upward from the pendulum’s rest position, and normalize the potential energy in such a way that it is zero if θ = π/2. Now the potential and kinetic energies can be written as P = −m g l cos θ

and

1 K = m l 2 θ˙ 2

(1.104)

respectively, and the total energy is just E = P + K = const. Choosing the actionangle variables as q = θ and p = m l 2 θ˙ (1.105) we get E=

p2 − m g l cos q ≡ H (q, p). 2 m l 2 q˙

(1.106)

Indeed, q˙ =

∂H p = ∂p m l2

and p˙ = −

∂H = −m g l sin q, ∂q

(1.107)

and accordingly dH ∂H ∂H = p˙ + q˙ dt ∂p ∂q p p = (−m g l sin q) + (m g l sin q) 2 ≡ 0, 2 ml ml

(1.108)

i.e. H = const. Notice that q and p describe the position of the oscillating mass and its velocity respectively, which justifies the terminology action-angle variables. Lagrangian and Hamiltonian dynamics are, of course, equivalent. But the advantages of Hamiltonian formalism are well known: specific features of a physical system are mostly not essential; all versions of the perturbation theory are simplified and standardized; its evolution equation has universal form in canonical variables related by a transformation of Fourier type with physical variables, etc. In particular, the Hamiltonian is the same for waves of various physical natures, with all specifics absorbed by the coupling coefficients and by the dispersion function of the linear modes. Later, we follow [260] in exposition and also in the notation for canonical variables, ak and ak∗ , because the Hamiltonian equations are most conveniently written in Fourier space (mostly referred to as k-space in physical texts) as i

∂H dak = ∗. dt ∂ak

(1.109)

1.3 Hamiltonian formalism

21

 The Hamiltonian function H = H ak , ak∗ (or just Hamiltonian), can usually, though not necessarily, be regarded as the total energy of the wave system. Indeed, H is equal to the sum of potential and kinetic energy in conservative systems, although in general Hamiltonian systems need not be conservative. Now to introduce a small parameter 0 < ε  1, we can just assume that wave amplitudes are small (e.g., when the elevation of gravity waves on the water surface is much smaller than the wavelength). Then the Hamiltonian can be expanded in the powers of ak and ak∗ : H = H2 + Hint , Hint = H3 + H4 + H5 + · · · ,

(1.110) (1.111)

where Hj is a term proportional to the product of j amplitudes ak , and the interaction Hamiltonian Hint describes dynamics, as explained below. Expansion (1.110) is only valid for small wave amplitudes, which generally speaking yield H3 > H4 > H5 > · · · .

(1.112)

However, in particular cases (e.g. for spin waves in magnetics with exchange interactions or Kelvin waves on quantum vortex lines), the odd expansion terms vanish. In these cases, instead of (1.112) we have H3 = H5 = H7 = · · · = 0

H4 > H6 > H8 > · · · .

(1.113)

Therefore, as a rule, three-wave interactions dominate in wave systems with small nonlinearity, e.g. for planetary waves in the atmosphere and oceans, capillary waves on the water surface, drift waves in plasmas, etc. This corresponds to the term ϕ 2 in (1.49). On the other hand, if resonance conditions (1.73) have no solutions or its coupling coefficients, as in (1.101), are equal to zero, then the cubic Hamiltonian vanishes, H3 = 0, and the leading nonlinear processes are four-wave interactions, described by quadric Hamiltonian H4 , and so on. Most physical phenomena can be described by three- or four-wave interactions though physically relevant examples with leading five- and even six-wave processes can be given (e.g. onedimensional gravity water waves [62] and Kelvin waves on quantum vortex lines [137] correspondingly).

22

Exposition

Dynamical equations The quadratic Hamiltonian H2 =

∞ 

ωk |ak |2 ,

(1.114)

n=1

according to equation (1.109) produces a linear equation of motion, i

dak = ωk a k , dt

(1.115)

and thus describes linear waves with the dispersion function ωk ≡ ω(k). The ∗ as they are Hamitonian H2 in (1.114) does not include terms ak a−k and ak∗ a−k removed by standard linear canonical transformation, which yields the diagonal form of (1.114). The first contribution to the interaction Hamiltonian Hint is given by the cubic Hamiltonian  1 ∗ H3 = V23 a1 a2 a3 123 + c.c., (1.116) k 1 ,k 2 ,k 3

which describes the processes of decaying of a single wave into two waves or confluence of two waves into a single one. In (1.116), notations aj ≡ akj are intro1 =V1 . duced and 123 ≡ (k 1 − k 2 − k 3 ) is the Kronecker symbol. Obviously, V23 32 ∗ Generally speaking, H3 also includes terms of the form a1 a2 a3 and a1 a2∗ a3∗ , but they can be eliminated by the corresponding nonlinear transformation that leads to the canonical form of cubic Hamiltonian H3 given by (1.116). Hamiltonian H2 + H3 with equation (1.109) yields the three-wave equation: i

  dak k 1∗ = ωk a k + a1 a2 k12 + Vk2 a1 a2∗ 1k2 . V12 dt

(1.117)

k 1 ,k 2

The canonical part of the quadric Hamiltonian reads H4 =



12 ∗ ∗ T34 a1 a2 a3 a4 12 34 ,

(1.118)

k 1 ,k 2 ,k 3 ,k 4

and describes four-wave scattering processes 2 ⇔ 2. The terms of the form a1 a2 a3 a4∗ , a1 a2 a3 a4 and their complex conjugates, describing 1 ⇔ 3 and 4 ⇔ 0 processes, can be eliminated by an appropriate nonlinear canonical transformation. This means, in particular, that only resonances of the form

1.3 Hamiltonian formalism

ω1 + ω2 = ω3 + ω4

23

(1.119)

have to be regarded, not those of the form ω1 + ω2 + ω3 = ω4

ω1 + ω2 + ω3 + ω4 = 0.

or

(1.120)

After that the quadric Hamiltonian takes the canonical form (1.118). Note that besides trivial symmetries with respect to indexes permutations, 1 ↔ 2 and 3 ↔ 4,

∗ 34 12 the coupling coefficient has the symmetry T12 = T34 , because the Hamiltonian has to be real, H4 = H4∗ . Now, the dynamical equation for the four-wave case follows from (1.109) with the Hamiltonian H = H2 + H4 : i

 dak k1 ∗ = ωk a k + T23 a1 a2 a3 k1 23 . dt

(1.121)

k 1 ,k 2 ,k 3

Terms on the right-hand side of (1.117) have time dependence of the form exp[−i(ω2 + ω3 )t] and exp[−i(ω2 − ω3 )t] respectively. Similar to the case of a driven pendulum, they become resonant if their frequencies coincide with the eigenfrequency of ak , ωk : ω2 + ω3 = ωk

or

ω2 − ω 3 = ω k .

(1.122)

Renaming indexes and taking into account Kronecker symbols in (1.117), resonance conditions for three-waves turn into ω1 + ω 2 = ω3 ,

k1 + k2 = k3

(1.123)

so that the first of these equations coincides with (1.73). Similarly, resonance conditions for four-wave interactions take the form ω1 + ω 2 = ω 3 + ω 4 ,

k 1 + k 2 = k3 + k4 .

(1.124)

Notice that the definition of resonance conditions in physics consists of two equations, while in mathematics it includes only the first of them. The simplest dynamical system for study, in the case of three-wave resonances, is a truncated system of three modes, also called a triad. A dynamical system corresponding to a triad can easily be deduced from (1.117): ⎧ 3 ∗B ∗B , ⎪ i B˙ = V12 ⎪ 2 3 ⎨ 1 ∗ ∗ 3 (1.125) i B˙ 2 = V12 B1 B3 , ⎪ ⎪ ⎩ ˙ 3B B , i B3 = V12 1 2

24

Exposition

where we introduce notation Bj ≡ aj exp(iωj t) for “slow” amplitudes of threewave resonances with resonance conditions (1.123). Analogously, for the case of four-wave resonances, the simplest dynamical system is truncated to four modes, called a quartet, and reads ⎧ 12 B ∗ B B , i B˙ 1 = T34 ⎪ ⎪ 2 3 4 ⎪ ⎪ ⎨i B˙ = T 12 B ∗ B B , 2 34 1 3 4 (1.126) 12 ∗ B ∗ B B , ⎪ ˙ = T i B ⎪ 3 1 2 34 4 ⎪ ⎪ ⎩ ˙ 12 ∗ B ∗ B B . i B4 = T34 3 1 2 To be more precise, we have to write ⎧ 12 B ∗ B B + (ω ˜ 1 − ω1 )B1 , i B˙ 1 = T34 ⎪ 2 3 4 ⎪ ⎪ ⎪ ⎨i B˙ = T 12 B ∗ B B + (ω˜ − ω )B , 2 3 4 2 2 2 34

1

12 ∗ B ∗ B B + (ω ⎪ ˜ 3 − ω3 )B3 , i B˙ 3 = T34 ⎪ 4 1 2 ⎪ ⎪ ⎩ ˙ ∗ 12 B ∗ B B + (ω ˜ 4 − ω4 )B4 , i B4 = T34 3 1 2

(1.127)

instead of (1.126), where ω˜ j − ωj =

4 

Tij |Bj |2 −

i=1

1 Tjj |Bj |2 , 2

(1.128)

are so-called “Stock’s-corrected” frequencies [223]. However, these additional terms are usually omitted in the literature, while suitable renormalization of ω-s yields the more usual form (1.126), without terms (ω˜ j − ωj )Bj included (e.g. [149], for one-dimensional quartets). In other cases they should be taken into account. In most cases, the systems (1.125) and (1.126) can be rewritten as ⎧ ∗ ˙ ⎪ ⎨B1 = ZB2 B3 , (1.129) B˙ 2 = ZB1∗ B3 , ⎪ ⎩˙ B3 = −ZB1 B2 , and

with real coefficient Z.

⎧ ⎪ B˙ 1 ⎪ ⎪ ⎪ ⎨B˙ 2 ˙ ⎪ B3 ⎪ ⎪ ⎪ ⎩˙ B4

= ZB2∗ B3 B4 , = ZB1∗ B3 B4 ,

= −ZB4∗ B1 B2 , = −ZB3∗ B1 B2 ,

(1.130)

1.3 Hamiltonian formalism

25

Definition 6. System (1.129) is called a primary dynamical system in a three-wave resonance system. System (1.130) is called a primary dynamical system in a fourwave resonance system. Primary dynamical systems describe time evolution according to the criterion of decay instability (see Section 4.2). However, in some physical systems, (1.125) and 3 and T 12 , which correspond (1.126) might have the same sign as the coefficients V12 34 to the explosive instability with a completely different time evolution. This subject is outside of the scope of the present book; however, the main results will be briefly outlined in Section 4.6, for the completeness of exposition. Primary systems play a major role in the dynamics of wave systems possessing nonlinear resonances. Similar to Lego pieces used by a child to build a house, we will construct dynamical systems for an arbitrary finite number of resonant waves using (1.125) or (1.126) as Lego pieces, depending on whether three- or four-wave resonances are relevant. Correspondingly, triads and quartets are called primary clusters while computing the resonance clustering, which will be done in Chapter 3. Examples In the Table 1.1, we present a few examples of physically relevant dispersion functions and coupling coefficients for various wave systems, allowing threeor four-wave resonances. All examples are given for two-dimensional wavevectors, notations are as follows: k = (m, n) and k = |k| are wavevector and its norm respectively; all other notations are some physical constants: g is the gravity acceleration, σ is the surface tension, ρ is the fluid density, E is the rotation frequency of the Earth, β the meridional gradient of the Coriolis parameter, ρ˜ is the Rossby deformation radius.

The explicit forms of both the dispersion function and the coupling coefficients depend, of course, on the choice of boundary conditions and variables. Indeed, the barotropic vorticity equation (1.78) regarded on an infinite β-plane with periodic boundary conditions in a square domain has dispersion function ω=

β ρ˜ 2 m 1 + ρ˜ 2 (m2 + n2 )

(1.131)

and the coupling coefficient can be found in Table 1.1, third row, in canonical variables.

26

Exposition Table 1.1 Examples of coupling coefficients for various wavesystems

σ k 3/2

3 V12 [259]

√ i √ ω1 ω2 ω3 8π 2σ

×

 K k2 ,k3

√ k1 k2 k3

−

Kk1 ,−k2 √ k3 k1 k2

−

Kk1 ,−k3 √ k2 k1 k3



where Kk2 ,k3 = (k2 · k3 ) + k2 k3 . (g k + σ k 3 )1/2

3 V12 [167]

2

ω2 −ω2 ω3 +ω32 k1 ω1

− ω2 k2 + ω3 k3

+ (k2 · k3 ) ωk22 −

ω3 k3

iβ √ 4π |m1 m2 m3 | ×

β ρ˜ 2 m 1+ρ˜ 2 k 2

3 V12 [200]

−

gk 1/2

12 T34 [256]

1 16π 2 (k1 k2 k3 k4 )1/4



− 

ω2 ω3 k1

ω1 k2 k3

n1 1+ρ˜ 2 k12

−

n2 1+ρ˜ 2 k22

−

n3 1+ρ˜ 2 k32

−12k1 k2 k3 k4

2) − 2 (ω1 +ω [ω3 ω4 (k1 · k2 − k1 k2 ) + ω1 ω2 (k3 · k4 − k3 k4 )] g2 2

3) − 2 (ω1 −ω [ω2 ω4 (k1 · k3 + k1 k3 ) + ω1 ω3 (k2 · k4 + k2 k4 )] g2 2

4) − 2 (ω1 −ω [ω2 ω3 (k1 · k4 + k1 k4 ) + ω1 ω4 (k2 · k3 + k2 k3 )] g2 2

1 k2 )(−k2 ·k4 +k2 k4 ) + 4(ω1 + ω2 )2 (k1 ·k2 −k 2 2

ω1+2 −(ω1 +ω2 )

1 k3 )(k2 ·k4 +k2 k4 ) + 4(ω1 − ω3 )2 (k1 ·k3 +k 2 2

ω1−3 −(ω1 −ω3 )

1 k4 )(k2 ·k3 +k2 k3 ) + 4(ω1 − ω4 )2 (k1 ·k4 +k 2 2

ω1−4 −(ω1 −ω4 )

+ (k1 · k2 + k1 k2 )(k3 · k4 + k3 k4 ) + (−k1 · k3 + k1 k3 )(−k2 · k4 + k2 k4 )  + (−k1 · k4 + k1 k4 )(−k2 · k3 + k2 k3 ) k2

12 T34 [177]

1

As was mentioned above, this same equation with zero boundary conditions on a sphere describes atmospheric planetary waves. Computed for physical variables, dispersion function, and coupling coefficients read

ω=

−2mE n(n + 1)

(1.132)

1.3 Hamiltonian formalism

27

and −2iZ(N2 − N3 )/N1 ,

−2iZ(N3 − N1 )/N2 ,

2iZ(N2 − N1 )/N3 ,

(1.133)

respectively, with Z given by (1.96). With zero boundary conditions in a square domain [L × L], the barotropic vorticity equation describes ocean planetary waves with dispersion function ω=

βL . √ 2π m2 + n2

(1.134)

An unfortunate attempt to compute the coupling coefficients for this case by hand can be found in [128]; cumbersome formulas obtained there were shown to be incorrect, by computing their magnitudes in Mathematica [127]. The correctness of the program was checked on the coupling coefficients for atmospheric planetary waves, whose magnitudes were compared with those given by analytical formulas (1.96), (1.133). The advantage of computing coupling coefficients in physical variables is as follows. The procedure, though elaborate, is straightforward and can be programmed in some symbolical language, for instance in Mathematica as was done in [127]. Still, explicit formulas for the coefficients have not been obtained, only the numerical magnitudes of the coupling coefficients, computed for a given solution of the corresponding resonance conditions. The problem is due to the fact that Mathematica sometimes does not seem to take care of special cases and consequently has problems with evaluating expressions depending on symbolic parameters. For instance, although 

2π

sin(mx) sin(nx)dx = π δm,n

(1.135)

0

for arbitrary integers m and n, in Mathematica Integrate[Sin[m*x]Sin[n*x], {x,0,2π }, Assumptions → m∈Integers && n∈Integers] yields 0 independently of m, n. For more discussion, see [127]. A combination of symbolical programming in Mathematica and classical hand rewriting in places where Mathematica fails yields awesome formulas partly shown below (M. Bustamante, 2009, private communication). The computations have been performed for the barotropic vorticity equation ∂ψ ∂ψ +β = −εJ (ψ, ψ) ∂t ∂x

(1.136)

28

Exposition

with zero boundary conditions x = [0, Lx ]; y = [0, Ly ]. The dynamical system for the slowly changing amplitudes of three resonance modes, written in physical variables, turns out to be of the form ⎧

2 m2 + n2 A ˙ 1 (t) = α1 A∗ A3 , ⎪ π ⎪ 1 1 2 ⎨ 2

2 2 (1.137) π m2 + n2 A˙ 2 (t) = α2 A∗1 A3 , ⎪ ⎪

⎩ 2 2 π m3 + n23 A˙ 3 (t) = α3 A1 A2 with coefficients α1 , α2 , α3 being functions of wavenumbers mj , nj . The explicit form of α1 is (partly) shown below. Notations E and I are for exponent exp and imaginary unit i, all other notations are self-explanatory. α1 =

1 × 16

(E^(-I (m1 + m2 + m3 + Sqrt[m1^2 + n1^2] + Sqrt[m2^2 + n2^2] - Sqrt[m3^2 + n3^2]) \[Pi]) ((I (m2^2 m3 n2 - 2 m3^3 n2 + m3 n2^3 + m2 m3 n2 Sqrt[m2^2 + n2^2] + 2 m2^3 n3 m2 m3^2 n3 + 2 m2 n2^2 n3 + 2 m2^2 Sqrt[m2^2 + n2^2] n3 - m3^2 Sqrt[m2^2 + n2^2] n3 + n2^2 Sqrt[m2^2 + n2^2] n3 - 2 m3 n2 n3^2 m2 n3^3 - Sqrt[m2^2 + n2^2] n3^3 + m2^2 n2 Sqrt[m3^2 + n3^2] - 2 m3^2 n2 Sqrt[m3^2 + n3^2] + n2^3 Sqrt[m3^2 + n3^2] + m2 n2 Sqrt[m2^2 + n2^2] Sqrt[m3^2 + n3^2] m2 m3 n3 Sqrt[m3^2 + n3^2] m3 Sqrt[m2^2 + n2^2] n3 Sqrt[m3^2 + n3^2] n2 n3^2 Sqrt[m3^2 + n3^2]))/(m1 + m2 - m3 + Sqrt[m1^2 + n1^2] + Sqrt[m2^2 + n2^2]- Sqrt[m3^2 + n3^2]) + (I (m2^2 m3 n2 2 m3^3 n2 + m3 n2^3 + m2 m3 n2 Sqrt[m2^2 + n2^2](...)

Twenty more pages, needed to accomplish this formula and denoted as “(. . . )” at the end of the expression, are omitted. More work is necessary to make these formulas usable analytically; another possibility to practically use them is as an on-line implemented service (see Appendix). The form of canonical coupling coefficients is usually more compact and easy to use; some examples are displayed in Table 1.1. In the 1st, 2nd, and 3rd columns respectively, the form of dispersion function for periodical boundary conditions, the standard notation for the coupling coefficients and the original reference, and

1.3 Hamiltonian formalism

29

the explicit form of coupling coefficients are given (for dynamical systems in canonical variables). These examples have been chosen for their simplicity; specific forms of coupling coefficients can be found in numerous papers. For instance, in [220] coupling coefficients are presented for five different types of plasma waves possessing three-wave resonances: electrostatic waves in an unmagnetized plasma, cold magnetized plasma, MHD waves, hot magnetized plasma, and electrostatic waves in a plasma cylinder (three-wave interactions). In [139] coupling coefficients are computed for four- and five-wave interactions among gravity water waves, etc. The problem, however, is then to find canonical variables for a given nonlinear PDE, specially for a PDE with a noncanonical Poisson structure, as is the case with the barotropic vorticity equation describing, in particular, planetary waves in the ocean (see, for instance, [200] for the construction and discussion). The importance of coupling coefficients is due to the fact that both qualitative and quantitative dynamics of a system possessing resonances depend on its magnitude. In particular, the dynamical system for three-wave resonance (1.125) is integrable in Jacobian elliptic functions and has periods of energy oscillations within the modes of a triad that are inversely proportional to Z. On the other hand, the dynamics of a cluster formed by two connected triads can be integrable or not, depending on the ratio of the corresponding coupling coefficients. Last, but not least, as our main goal in this book is to demonstrate that nonlinear resonance analysis is a constructive tool, allowing us to construct explicitly dynamical systems corresponding to resonance clusters, we give in Table 1.1 the known expressions for coupling coefficients for some types of waves. Each coefficient is accompanied by the reference for the following reason. In various publications, definitions of the Hamiltonian expansion differ by a coefficient 2 in front of the Hamiltonian, even in works by the same author – compare e.g. [255], (1.11)– (1.12) with [260], (1.2.41b). This is an important fact to know when coming back to physical variables. Remark 1.1. In Definition 4, we followed Whitham [248] who regarded a dispersion as a real-valued function. However, in some publications, dispersion is regarded as a complex-valued function (see [51] and bibliography therein). In this case, the resonance conditions (1.73) have to be rewritten as Re(ω1 + ω2 ) = Re(ω3 ),

k 1 + k 2 = k3 ,

(1.138)

where the notation Re(ω) is used for the real part of the dispersion function. Similarly we can rewrite resonance conditions for s, s ≤ ∞ waves. This simple operation does not influence the results presented in the following chapters; we have chosen to regard a dispersion as a real-valued function for simplicity of presentation.

2 Kinematics: Wavenumbers

All the same it is desirable, citizen artiste, that you expose the technique of your tricks to the spectators without delay. M. Bulgakov Master and Margarita

2.1 An easy start Let us now consider the simplest case of resonance conditions ! ω1 + ω 2 = ω3 , k1 + k 2 = k 3 ,

(2.1)

and try to find integer solutions of (2.1) for some specific dispersion function, say ω ∼ (m2 + n2 )1/2 (sound waves). Sign ∼ in this and all other dispersion functions that will be studied further means that constant coefficients of proportionality are omitted, for obvious reasons. If we do not have any idea about simplification and/or solving the system (2.1) analytically, our only hope is the trivial though very general problem-solving technique called brute-force or exhaustive search. To use it, we have first to transform System (2.1) into the Diophantine form ⎧



2 2 1/2 + m2 + n2 1/2 = m2 + n2 1/2 , ⎪ ⎨ m1 + n1 2 2 3 3 m1 + m2 = m3 , ⇒ ⎪ ⎩ n1 + n2 = n3 , ⎧

2





2 2 2 1/2 m2 + n2 1/2 + m2 + n2 = m2 + n2 , ⎪ ⎨ m1 + n1 + 2 m1 + n1 2 2 2 2 3 3 m1 + m2 = m3 , ⎪ ⎩ n1 + n2 = n3 , 30

(2.2)

(2.3)

2.1 An easy start

⇒ ⎧





2

2

 2 2 2 2 2 2 2 2 2 ⎪ ⎨4 m1 + n1 m2 + n2 = m3 + n3 − m1 + n1 − m2 + n2 , m1 + m2 = m3 , ⎪ ⎩ n1 + n2 = n3 .

31

(2.4)

Now we have a polynomial system of equations with (a) two linear equations, each in three variables, presenting no computational difficulties, and (b) one polynomial in six variables, with the maximal cumulative degree of its monomials being 4. This amounts to, even in the relatively small spectral domain of wavenumbers, say m, n ≤ 103 , manipulating with integers of the order of 1012 . At first glance, it does not seem a problem for a physicist, because probably the most often encountered context in which a physicist uses big natural numbers is in the generation of random numbers. But generation of a random number of order 1019 is much faster than establishing that a number of order 108 can be decomposed into the sum of two integer squares. Computational problems with integers present some specific challenges. First of all, the solution must be precise and not approximate as with “reals” (i.e. floatingpoint numbers). Consider a circle of some astronomic radius, say R = 10100. Its area can be computed in microseconds with any reasonable precision. However, calculating the precise number of integer points within that same circle by computer is an unrealistic task for modern means – exhaustive search for multivariate problems in integers consumes exponentially more time with each variable and size of the domain to be explored. All these reasons make the need for effective algorithms unavoidable and some of them – based on so-called q-class decomposition – are presented below in this chapter. The main idea of the q-class decomposition is based on two almost obvious facts from school mathematics: √ √ r = γ q with integer √ √ γ , q and such that q is square-free (e.g. 162 = 9 2, γ = 9, q = 2); √ √ • equation a q1 + b q2 = 0 has no nontrivial solutions with rational a, b if q1 = q2 √ √ (e.g. a 5 + b 17 yields a = b = 0). • for arbitrary integer r, the following presentation is unique:

These statements can be generalized for the case of the arbitrary noninteger rational power of r in the first statement and any finite number of terms in the second, yielding the very powerful computation methods presented below. In particular, they mean that (2.1) has integer solutions only among wavevectors with the same q. Now, q-class decomposition means simply that we first divide all wavevectors into classes with the same q and then look for solutions only among wavevectors belonging to one class.

32

Kinematics: Wavenumbers

Another great benefit of q-class decomposition is the following. Sometimes, when the first equation of (2.1) is rewritten in its q-class form, it turns into a Diophantine equation with known properties and/or solutions, and numerical simulations are not necessary. The most fascinating example of this kind is presented in Section 2.2, where it is shown that there are no three-wave resonances among irrotational capillary waves in rectangular domains because frequency resonance turns into a particular case of Fermat’s last theorem. In Chapter 2, we present both analytical and numerical methods for solving the resonance conditions with s = 3 and s = 4, for a wide class of dispersion functions. 2.2 Irrational dispersion function, analytical results Let us regard an irrational dispersion function of some general form and define q-classes. For a given c ∈ N, c = 0, 1, −1, consider the set of algebraic numbers Rc = ±k 1/c , k ∈ N. Any such number kc has a unique representation kc = γ q 1/c ,

γ ∈ Z,

(2.5)

where q is a product q = p1e1 p2e2 . . . pnen ,

(2.6)

while p1 , . . . , pn are all different primes and the powers e1 , . . . , en ∈ N are all smaller than c. Definition 7. The set of numbers from Rc having the same q is called q-class, or class of index q, and is denoted as Clq . For a number k(c) = γ q 1/c , γ is called the weight of k(c) . The following two lemmas are easily obtained from elementary properties of algebraic numbers. Lemma 1. For any two numbers k1 , k2 belonging to the same q-class, all their linear combinations (2.7) a1 k1 + a2 k2 , a1 , a2 ∈ Z with integer coefficients belong to the same class q. Indeed, if k1 = γ1 q 1/c , k2 = γ2 q 1/c , then 1/c

a1 k1 + a2 k2 = γ1 q1

1/c

+ γ2 q 2

= (γ1 + γ2 )q 1/c .

(2.8)

In other words, every class is a one-dimensional module over the ring of integers Z.

2.2 Irrational dispersion function, analytical results

33

Example 5: Case c = 2 √ √ √ Numbers k1 = 8 and k2 = 18 belong to the same class 2: k1 = 2 2, k2 = √ 3 2. According to√Lemma √ 1, their sum k1 + k2 belongs to the same class, and, indeed, k1 + k2 = 50 = 5 2. Lemma 2. For any n numbers k1 , k2 . . . kn belonging to pairwise different q-classes, the equation k1 ± k2 ± · · · ± kn = 0 (2.9) has no nontrivial solutions. The statement of Lemma 2 follows from some known properties of algebraic numbers and we are not going to present a detailed proof here. The general idea of the proof is very simple and was formulated in the previous section for a linear combination of two different irrational numbers. As for the general situation, we can apply the Besikovitch theorem [19]. Theorem 2 (Besikovitch theorem). Let a1 = b1 p1 ,

a2 = b2 p2 , . . . ,

as = bs ps ,

(2.10)

where p1 , p2 , . . . , ps are different primes, and b1 , b2 , . . . , bs are positive integers not divisible by any of these primes. If x1 , x2 , . . . , xs are positive real roots of the equations x n1 − a1 = 0,

x n2 − a2 = 0, . . . ,

x ns − as = 0,

(2.11)

and P (x1 , x2 , . . . , xs ) is a polynomial with rational coefficients of degree less than or equal to n1 − 1 with respect to x1 , less than or equal to n2 − 1 with respect to x2 , and so on, then P (x1 , x2 , . . . , xs ) can vanish only if all of its coefficients vanish. Corollary 1. Any equation a1 k1 + a2 k2 + · · · + an kn = 0, ai ∈ Z

(2.12)

with k1 , k2 , . . . , kn belonging to pairwise different q-classes has no nontrivial solutions. This theorem was first used in the context of nonlinear resonances in [111]. 1/c Obviously, while ki can be represented as γi qi , each number 1/c

ai ki = sgn(ai )|ai |γi qi .

(2.13)

34

Kinematics: Wavenumbers

Example 6: Case c = 2 √ √ √ Numbers k = 8 and k = 12 belong to different classes: k = 2 2, k2 = 1 2 1 √ 2 3 and, according to Lemma 2, the equation a1 k1 + a2 k2 = 0 has only the trivial solution a1 = a2 = 0. The role of classes in the study of integer solutions of (2.1) lies in the following theorem. Theorem 3. Consider the equation a1 k1 + a2 k2 + · · · + an kn = 0,

ai ∈ Z ,

1/c

where each ki = γi qi belongs to some class qi ∈ q1 , q2 . . . ql , equation is equivalent to the system ⎧ ⎪ aq1 ,1 γq1 ,1 + aq1 ,2 γq1 ,2 + · · · + aq1 ,n1 γq1 ,n1 = 0 ⎪ ⎪ ⎪ ⎪ ⎨a γ +a γ + ··· + a γ =0 q2 ,1 q2 ,1

q2 ,2 q2 ,2

(2.14) l < n. This

q2 ,n2 q2 ,n2

⎪ ... ⎪ ⎪ ⎪ ⎪ ⎩a γ ql ,1 ql ,1 + aql ,2 γql ,2 + · · · + aql ,nl γql ,nl = 0.

(2.15)

Let us rewrite the initial equation grouping numbers belonging to the same class as follows: (kq1 ,1 + kq1 ,2 + · · · + kq1 ,n1 ) + (kq2 ,1 + kq2 ,2 + · · · + kq2 ,n2 ) + · · · + (kql ,1 + kql ,2 + · · · + kql ,nl ) = 0,

(2.16)

so that all numbers in the ith bracket belong to the same class qi and all qi are different. From Lemma 1, it follows that each sum kqi ,1 + kqi ,2 + · · · + kqi ,ni is some number kqi of the same class qi and the equation can be re-written as kq1 + kq2 + · · · kql = 0

(2.17)

with all different qi . It immediately follows from Lemma 2 that kq1 = 0, kq2 = 0, · · · , kql = 0. Example 7: Simplification of a five-term equation Consider the equation √ √ √ √ √ a1 8 + a2 12 + a3 18 + a4 24 + a5 48 = 0. This is equivalent to

(2.18)

2.2 Irrational dispersion function, analytical results

√ √ √ √ √ 2a1 2 + 2a2 3 + 3a3 2 + 2a4 6 + 4a5 3 = 0,

35

(2.19)

and according to the Besikovitch theorem is equivalent to the system ⎧ ⎪ ⎨2a1 + 3a3 = 0, 2a2 + 4a5 = 0, ⎪ ⎩ a4 = 0,

(2.20)

which is in every respect much simpler than the original equation. All these considerations are of utmost importance for the development of fast computing algorithms for solving equations for resonance conditions. Some examples of this approach discussed below are chosen due to their relevance for physical applications. Principal example: ω ∼ (m2 + n2 )−1/2 (ocean planetary waves) As our principal example here, we follow [110] and take the dispersion function for ocean planetary waves in the square domain with zero boundary conditions, which yields resonance conditions of the form "

1

1 1 =" ±" , m23 + n23 m21 + n21 m22 + n22 n3 = n1 ± n 2 .

(2.21)

(2.22)

For arbitrary wavevector k = (m, n), m, n ∈ N, to construct q-classes (see Definition 7), we use representation |k| =



√ m2 + n2 = γ q,

γ, q ∈ N

(2.23)

with square-free q, i.e. q = p1 p2 . . . ps , and all pj are different primes for j = 1, 2, . . . , s. Lemma 3. If a solution of (2.21) is composed of three vectors (mi , ni ), i = 1, 2, 3, then they belong to the same q-class, i.e. ∃q ∈ N : (mi , ni ) ∈ Clq ,

i = 1, 2, 3.

(2.24)

The statement of this Lemma is equivalent to irrationality of the square root of a product of different primes.

36

Kinematics: Wavenumbers

Some general properties of classes are given by Lemma 4. Lemma 4. The following properties of q-classes hold: (1) Clq1 ∩ Clq2 = {0} ⇔ Clq1 = Clq2 (intersection of two classes is not empty iff these classes coincide); (2) card {Clq } = ∞ (there exists an infinite number of classes); (3) Clq = {0} ⇒ card Clq = ∞ (every nonempty class consists of an infinite number of elements); (4) Clq = {∅} ⇔ q = p1 p2 · · · pn , where pi ∈ P are different primes such that pi ≡ 3 (mod 4); (5) Clq = {0} & q > 1 ⇒ q = a 2 + b2 for some integers a, b ∈ Z (in each nonempty class, the minimal element has norm q); (6) Cl1 = {m, n : m2 + n2 = k 2 , m = a 2 − b2 , n = 2ab, k = a 2 + b2 , a > b} (elements of the class Cl1 can be parameterized by two parameters).

Indeed, (1) follows from definition, (2) follows from the fact that the set of primes is infinite, (3) is due to the fact that vectors (m, n) and (sm, sn) have the same index for arbitrary s ∈ N. In order to prove (4) and (5), we should use the Euler theorem. Theorem 4 (Euler’s theorem on a two-squares presentation of an integer). Natural number N, N = p1α1 p2α2 · · · pnαn can be presented as a sum of two squares iff {pi ∈ 4N + 3 ⇒ αi ∈ 2N}. The last property (6) follows from known parametrization of Pythagorean numbers. Complete solution of (2.21) Let us rewrite (2.21) as 1 1 1 = + , x3 x2 x1

x i ∈ N,

∀i = 1, 2, 3.

(2.25)

Now x2 x1 = x3 (x2 + x1 ) and the introduction of new variables x = x2 /x3 , y = x1 /x3 leads to xy = x + y. Setting y = tx with some t ∈ Q, we get tx 2 = x + tx ⇒ tx = 1 + t,

x=

1+t , t

y = 1 + t.

(2.26)

The rational number t has form t = r/s with some integers r, s, i.e. r +s x2 , = x3 r

x1 r +s (r + s) ⇒ x2 = x3 , = x3 s r

x1 = x3

(r + s) , s

(2.27)

which provides the complete set of solutions for (2.25). Indeed, let r, s ∈ N, (r, s) = 1, then the general form of solution is

2.2 Irrational dispersion function, analytical results

x1 = x˜1 r(r + s),

x2 = x˜1 s(r + s),

x3 = x˜1 rs, for some x˜1 ∈ N.

37

(2.28)

Obviously, the general form of solution does not depend on the q-class index and is the same for arbitrary q. To return to (2.21), let us fix some nonempty class Clq and consider all representations of numbers x12 q, x22 q, x32 q as sums of two squares (they exist due to the Euler theorem), then ⎧ 2 2 2 2 ⎪ ⎪m1,j + n1,j = x1 q = [x˜1 r(r + s)] q, ⎨ (2.29) m22,i + n22,i = x22 q = [x˜1 s(r + s)]2 q, ⎪ ⎪ ⎩ 2 m3,l + n23,l = x32 q = (x˜1 rs)2 q. The indexes i, j, l show that in each case there may exist several different decompositions into a sum of two squares. Jacobi’s two squares theorem [93] allows us to enumerate them. Theorem 5 (Jacobi’s two square theorem). The number of representations of a positive integer as a sum of two squares is equal to four times the difference of the numbers of divisors congruent to 1 and 3 modulo 4. Structural properties of q-classes More detailed studies of the q-classes for this case were carried out in [110]. In particular, gross and fine structures of q-classes are defined there – the first corresponds to general properties valid for arbitrary class index q, while the latter describes characteristics depending on the number of representations of q as a sum of two squares. Gross structures of all classes are always isomorphic and fine structures can be isomorphic under certain conditions; for instance, all classes with odd prime indices q have isomorphic fine structures. This means that for each class there exists an infinite number of isomorphic ones. However, on average, as the index grows, the density of isomorphic fine structures decreases, because large numbers on average factorize into a greater number of primes. Therefore, the following question naturally arises: What is the average number M(R) of vectors with equal norms in a fixed domain of radius R? The answer is given by the following theorem: Theorem 6 (on density of equal norms). The density evaluation is $ #√ √ π 2√ ln R M(R) = , ln R + O 4CG ln ln R where CG is a constant.

(2.30)

38

Kinematics: Wavenumbers

The proof makes use of two well-known results of the analytical number theory – the theorem on the distribution of primes in an arithmetic progression [55] and the Bredichin theorem ([26]). Notice, however, that the Bredichin theorem does not give a constructive method for the calculation of the group constant CG , which is a nontrivial task. A combination of analytical reasoning and numerical analysis allowed us to conclude that CG ≈ 0.78. Multiplicative properties of solutions Knowledge of the multiplicative properties of q-classes helps to shed some light on the interrelation between the number of resonances and the form of spectral domains. Indeed, for all dispersion functions of the form ω = ω(k), with k = |k|, k = (m, n) and a rectangular domain p × q, the two sides p and q of this domain participate in the function ω only as factors of the corresponding components of wavevectors with the same degree. Since p and q are integers, the  simple substitution m = m/q and n = n/p followed by substitutions m = m p  and n = n q show that the study of resonances in rectangular domains can be reduced to studying some special solutions in square domains. Namely, the solutions in square domains must satisfy certain divisibility properties for corresponding wavevector components, for instance m must be divisible by p and n must be divisible by q. This means, in particular, that the complete set of solutions can be divided into subsets of solutions modulo p and/or q, i.e. solutions in the fields Zp = Z/p Z, Zq =Z/q Z or their (possible) intersection. (Of course, (2.21) have to be transformed into the Diophantine form first.) This approach allows us to evaluate the number of wavevectors interacting resonantly, i.e. the density of resonances for basins with different ratios of sides. The use of classical results from [238] and [148] allows us to evaluate the number of basins where resonances are possible as follows:  Cx x ν(x) = √ O √ . (2.31) ln x ln x ln x This means that for all basins with sides √ p = 1, 2, . . . , x and (p, q) = 1, resonances are possible only for about Cx ln x of them, where C is a constant. For the overwhelming majority of basins, resonances are impossible. In the cases where resonances do exist, their number depends substantially on the form of the basin and the size of the spectral domain m, n. Other examples can be found in [107, 127]. For instance, in the spectral domain m, n ≤ 50, no solutions have been found with p/q = 11/29, while for p/q = 1/1 and p/q = 3/1, the number of solutions changes from 76 to 23.

2.2 Irrational dispersion function, analytical results

39

Notice that the second side of the basin participates only in the form of (p, q) = 1. This is due to a special feature of planetary waves: for them, we have n1 + n2 = n3 and no such condition for mi . In the case when both these linear equations are valid, solutions should be considered also in the fields Z/q1 Z, . . . , Z/qn Z,

(2.32)

which would reduce the number of basins in which resonances are possible still further (here we denote by q1 , . . . , qn all the prime factors of the side q). We have regarded this example of ocean planetary waves in detail because similar partitions into q-classes exist for a large number of dispersion functions, including quadratic irrationality. Below we regard briefly a few other examples. Example 8: ω ∼ (m2 + n2 + l 2 )−1/2 (internal waves) The dispersion function for internal waves can be taken in the form ω ∼ (m2 + n2 + l 2 )−1/2 . Correspondingly, resonance conditions for three internal waves in an exponentially stratified rotating liquid have the form ⎧ ⎪ " m1 + " m2 = " m3 , ⎪ ⎪ 2 +n2 +l 2 2 +n2 +l 2 ⎪ m m m23 +n23 +l32 ⎪ 1 1 1 2 2 2 ⎪ ⎪ ⎨m + m = m , 1 2 3 (2.33) ⎪ ⎪ n1 + n2 = n3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩l + l = l . 1 2 3 Again, the necessary conditions for three wavevectors ((m1 , n1 , l1 ), (m2 , n2 , l2 ), (m3 , n3 , l3 ))

(2.34)

to form a solution to equations (2.33) are analogous to our principal example and q-classes are constructed as follows: " √ m2i + n2i + li2 = ki q, i = 1, 2, 3 (2.35) with ki , q ∈ N, and q is square-free. We can proceed further as in the principal example: 1. All wavevectors (i.e. the set of nodes (m, n, l) of three-dimensional integer lattice) can be partitioned into disjoint classes Clq . 2. The first three statements (1)–(3) of Lemma 4 hold. 3. Statement (4) of Lemma 4 should be replaced by Clq = {0} ⇔ q = 8d − 1, which is a direct corollary of Gauss’ theorem:

d ∈ N,

(2.36)

40

Kinematics: Wavenumbers

Theorem 7 (Gauss’ theorem on a three squares representation of an integer). A positive integer is a sum of three squares iff it is not a number of the form 4a (8b − 1) with integer a, b. The general solution of the first two equations of System (2.33) can be found in a way quite similar to the principal example. Indeed, let us rewrite the first two equations as one equation m2 m1 + m2 m1 √ + √ = √ , k1 q k2 q k3 q

(2.37)

then



 m1 k1−1 − k3−1 = m2 k3−1 − k2−1 ⇒ m1 k1−1 (k3 − k1 ) = m2 k2−1 (k2 − k3 ).

(2.38)

Introducing new variables x = k3 /k1 , y = k3 /k2 , we get m1 (x − 1) = m2 (1 − y) and setting x˜ = x − 1, y˜ = y − 1 leads to xm ˜ 1 = −ym ˜ 2 . If we put x˜ = −t y, ˜ then t = p/r ∈ Q and x˜ =

p y, ˜ r

k3 = 1 + x, ˜ k1

x = 1 + x, ˜

y = 1 + y˜ =

k3 p + r x˜ k3 ⇒ k1 = , = k2 p 1 + x˜

p + r x˜ , ⇒ p k2 =

pk3 . p + r x˜

(2.39) (2.40)

The choice k3 = (1 + x)(p ˜ + r x)s, ˜ s = 1, 2, 3 . . . provides that all k1 , k2 , k3 are integers and the full set of solutions (probably with repetitions) of System (2.33) can be written as ⎧ 2 2 2 2 ⎪ ˜ 2 s 2 q, ⎪m1 + n1 + l1 = k1 q = (p + r x) ⎪ ⎨ m22 + n22 + l22 = k22 q = p 2 (1 + x) ˜ 2 s 2 q, ⎪ ⎪ ⎪ ⎩ 2 ˜ 2 (p + r x) ˜ 2 s 2 q. m3 + n23 + l32 = k32 q = (1 + x)

(2.41)

Keeping in mind that m3 = m1 + m2 ⇒ m3 = (r − p)m1 /r, m2 = −pm1 /r,

(2.42)

2.2 Irrational dispersion function, analytical results

41

let us take m1 = ra, a = 1, 2, 3 . . . in order to provide integer values for all mi . Finally ⎧ 2 2 2 ⎪ ˜ 2 s 2 q − r 2 a 2, ⎪n1 + l1 = k1 q = (p + r x) ⎪ ⎪ ⎪ ⎪ ⎪ ˜ 2 s 2 q − p2 a 2, n22 + l22 = k22 q = p 2 (1 + x) ⎪ ⎪ ⎪ ⎪ ⎨n2 + l 2 = k 2 q = (1 + x) ˜ 2 (p + r x) ˜ 2 s 2 q − (r − p)2 a 2 , 3 3 3 (2.43) ⎪ ⎪m1 = ra, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m2 = −pa, ⎪ ⎪ ⎪ ⎩ m3 = (r − p)a. Notice that the first three equations of System (2.43) are quite similar to System (2.29) and can be solved (ineffectively) by using the Euler theorem. In this example, we were able to include at least one linear equation for mi in the general formulas for solutions, but two others (for ni and li ) are still to be checked. Example 9: ω ∼ (m2 + n2 )3/4 (irrotational capillary waves) Capillary waves in the rectangular domain have the dispersion function ω2 = k 3 with k = |k| and System (2.1) reads as ⎧" " " ⎨ 3 k1 + k23 = k33 , (2.44) ⎩ k1 + k2 = k3 . Construction of q-classes Clq is similar to that in the previous section: ki = pi2 qi with square-free qi . Then all qi coincide and " " " 3 3 k1 + k2 = k33 ⇒ p13 + p23 = p33 . (2.45) This equation is a particular case of Fermat’s last theorem and therefore has no integer solutions. Example 10: ω ∼ (m2 + n2 )1/2 (sound waves) The dispersion function for sound waves has the form ω ∼ (m2 + n2 )1/2 . Correspondingly, resonance conditions for three sound waves have the form " " ⎧" 2 + n2 + m2 + n2 = m2 + n2 , ⎪ m ⎪ 1 1 2 2 3 3 ⎨ (2.46) m1 + m2 = m3 , ⎪ ⎪ ⎩ n1 + n2 = n3 , and q-class construction coincides with that given in Section 2.2.

42

Kinematics: Wavenumbers

It is easy to see that (2.46) has an infinite number of solutions, for instance: k1 = (m1 , n1 ),

k2 = (t1 m1 , t1 n1 ),

k3 = ((t1 + 1)m1 , (t1 + 1)n1 )

(2.47)

with some rational parameter t1 . Parameter t1 should be chosen in such a way that wavenumbers t1 m1 , t1 n1 , (t1 +1)m1 , (t1 +1)n1 are integers. It holds, for instance, if t1 itself is an integer. An interesting feature of this example is the following. In Section 2.2, we described all vectors belonging to the class Cl1 by a two-parametric series with some parameters a and b. This should not be understood that only prime Pythagorean triples ClP rime = {m, n : m2 + n2 = k 2 , m = a 2 − b2 , n = 2ab, k = a 2 + b2 , a > b} (2.48) with two natural parameters a and b are members of this class: only the inclusion ClP rime ⊂ Cl1 takes place. Moreover, it is easy to prove that in the case of sound waves ClP rime = ∅. Indeed, mi = ai2 − bi2 ,

and ni = 2ai bi ∀i = 1, 2, 3,

(2.49)

allows us to rewrite (2.46) as " " ⎧" 2 + b2 + a 2 + b2 = a 2 + b2 , ⎪ a ⎪ 1 2 2 3 3 ⎪ ⎨ 1 2 2 2 2 2 2 a1 − b1 + a2 − b2 = a3 − b3 , ⎪ ⎪ ⎪ ⎩ a1 b1 + a2 b2 = a3 b3 .

(2.50)

Since a, b are integers and a > b, we have a = pb with some rational p, p > 1 (p is not arbitrary, but at the moment it does not matter). Now (2.50) takes the form ⎧    ⎪ b1 p 2 + 1 + b2 p 2 + 1 = b3 p 2 + 1, ⎧ ⎪ ⎪ ⎨ b1 + b2 = b3 , ⎨ b12 (p 2 − 1) + b22 (p 2 − 1) = b32 (p 2 − 1), ⇒ ⎩b 2 + b 2 = b 2 . ⎪ ⎪ ⎪ 1 2 3 ⎩b2 p + b2 p = b2 p, 1 2 3

(2.51)

This yields b1 = 0 or b2 = 0, and correspondingly a1 = 0 or a2 = 0, i.e. class ClP rime is empty. On the other hand, it easy to check that three wavevectors (3, 4)(6, 8)(9, 12)

(2.52)

2.2 Irrational dispersion function, analytical results

43

do form a solution to (2.46) with all three vectors belonging to Cl1 . However, the parametrization of vectors (3, 4), (6, 8), (9, 12) yields √ √ √ √ a1 = 2, b1 = 1; a2 = 2 2, b2 = 2 2; a3 = 2 3, b3 = 3

(2.53)

correspondingly, where some parameters are not integers. Example 11: ω ∼ (m2 + n2 )1/4 (gravity surface waves in water) The dispersion function for gravity water waves has the form ω ∼ (m2 + n2 )1/4 . Correspondingly, resonance conditions for four waves have the form ⎧





2 2 1/4 + m2 + n2 1/4 = m2 + n2 1/4 + m2 + n2 1/4 , ⎪ ⎪ m1 + n1 2 2 3 3 4 4 ⎪ ⎨ m1 + m2 = m3 + m4 , ⎪ ⎪ ⎪ ⎩ n1 + n2 = n3 + n4 .

(2.54)

Here we define the q-class index as a number q such that (m2 + n2 )1/4 = kq 1/4 ,

k, q ∈ Z,

(2.55)

where q does not contain fourth degrees, i.e. α

q = p1 1 . . . psαs , αi ∈ {1, 2, 3},

(2.56)

where all pi are different primes, i = 1, 2, . . . , s. Now the analog of Lemma 4 for this case can be proved. It is evident that properties (1), (2), and (3) hold for these classes. It can also be shown directly that equation (2.54) has an infinite number of solutions. Indeed, let R be a natural number that can be partitioned into a sum of two squares in no less than four different ways (such a number is easy to obtain by using the Euler theorem mentioned in the principal example); it is sufficient to take as R the product of the corresponding number of prime factors of the form 4K + 1. Substitution of these partitions into (2.54) yields an identity R = m21 + n21 = m22 + n22 = m23 + n23 = m24 + n24 .

(2.57)

As the set of primes is infinite, it follows that there is an infinite number of such Rs and consequently that there is an infinite number of solutions of (2.54).

44

Kinematics: Wavenumbers

Parametric series of solutions 1. Trivial quartets. Quartets k1 = (a, b),

k2 = (c, d),

k3 = (a, b),

k4 = (c, d)

(2.58)

and k1 = (a, b),

k2 = (c, a −b +c),

k3 = (b, a),

k4 = (a −b +c, c)

(2.59)

with arbitrary integers a, b, c, d give its solution, as well as any proportional quartets corresponding to the multiplication of all wavenumbers by the same integer. 2. Tridents. A less trivial example for a trident quartet reads k1 = (a, 0),

k2 = (−b, 0),

k3 = (c, d),

k4 = (c, −d),

(2.60)

(the name is due to its form in the k-space [125]). We will discuss special geometrical properties of tridents, justifying their names, in Chapter 3. Two-parametric series of solutions also exists for this case: a = (s 2 + t 2 + st)2 ,

b = (s 2 + t 2 − st)2 ,

c = 2st (s 2 + t 2 ),

d = s4 − t 4 (2.61) with arbitrary integer s, t. This series gives solutions to (2.54), although not all of them. Parametrization (2.61) is constructed in such a way that norms of all four vectors are full squares, i.e. again vectors k1 , k2 , k3 , k4 belong to the same class Cl1 . 3. Three-parametric series. It is easy to see that an arbitrary wavevector (m, n) takes part in an infinite number of resonances if the spectral domain is unbounded. Indeed, let us fix m and n, then a quartet (m, n)(t, −n) → (m, −n)(t, n)

(2.62)

is an angle-resonance with arbitrary t = 0, ±1, ±2, . . . . Angle-resonances are resonances formed by two pairs of wavevectors with pairwise equal lengths; their special role in the dynamics of the wave system will be discussed in the next chapter. 4. Five-parametric series. More involved are the five-parametric series of angleresonances found from the following considerations. For angle-resonances of four wavevectors (a, b)(c, d) → (p, q)(l, m), (2.54) can be rewritten as ! a 2 + b2 = p 2 + q 2 , c2 + d 2 = l 2 + m2 , (2.63) a + c = p + l, b + d = q + m.

2.3 q-class decomposition

45

Simple algebraic transformations and known parameterizations of the sum of two integer squares (e.g. for a circle or for the Pythagorean triples) yield ⎧ ⎪ a = (s 2 − t 2 )/(s 2 + t 2 ), b = 2st/(s 2 + t 2 ), ⎪ ⎪ ⎨ (2.64) p = (f 2 − g 2 )/(f 2 + g 2 ), q = 2fg/(f 2 + g 2 ), ⎪ ⎪ ⎪ ⎩d = (a 2 + b2 + ac − ap − cp − bq)/(q − b). Analytical series are very helpful not only for computing resonant quartets and clusters structure, but also while investigating asymptotic behavior of coupling coefficients. 2.3 q-class decomposition The computational aspect of q-classes presented in the previous sections is an especially important illustration to the main idea of the algorithm we are going to present now. Indeed, suppose we were to find all the exact solutions to (2.18) in some finite domain 1 ≤ ai ≤ D. Then the straightforward iteration algorithm needs O(D 4 ) floating-point operations, ai = 1, . . . , D, i = 1, 2, 3, 4 (even ignoring difficulties with floating-point arithmetic precision for large D). On the other hand, solutions of System (2.20) can, evidently, be found in O(D) operations with integer numbers. To show the power of the approach outlined above in practice, we proceed as follows. First we give a detailed description of the algorithm that is used to find all solutions to resonance conditions for gravity surface waves in water in a finite domain. We also estimate computational complexity and memory requirements for its implementation and present the results of our computer simulations. In the next sections, we discuss reusability of this algorithm and transform it to solve a similar problem for three-wave resonances of ocean planetary waves and sound waves. Further, we briefly discuss the applicability of our algorithm to other types of waves. One q-class The procedure outlined in this section was first suggested in [117]. Resonance conditions for gravity surface waves in water (2.54) are our main object of study in this section: ⎧

1/4 2

1/4 2

1/4 2

1/4 ⎪ m21 + n21 + m2 + n22 = m3 + n23 + m4 + n24 , ⎪ ⎪ ⎨ (2.65) m1 + m2 = m3 + m4 , ⎪ ⎪ ⎪ ⎩ n1 + n2 = n3 + n4 .

46

Kinematics: Wavenumbers

We represent the solution as a quartet (m1L , n1L )(m2L , m2L ) ⇒ (m1R , n1R )(m2R , m2R )

(2.66)

to discern between the left and right sides of the equations. We are going to find all four-tuples such that −D ≤ mi , ni ≤ D, i = 1, 2, 3, 4, for some large enough D ∈ N. Our model domain is D = 103 . To solve numerically irrational equations in integers, two approaches are used. The first approach is to get rid of irrationalities. For a simple equation such as √ √ a x = b y, this approach is reasonable, but for System (2.65), containing four fourth-degree roots, this approach is out of the question. The second approach is to solve the equations using floating-point arithmetic precision, and to find some (domain dependent) lower estimate for the deviation in the approximate solutions, allowing us to arrive at exact solutions with the deviation due only to floating-point arithmetic precision. However, it is not as easy to find the corresponding estimate for (2.65), and anyway the computational complexity would be at least of order O(D 5 ). Our primary goal is to develop a generic algorithm, suitable for large domains and a wide class of wave types, also in more variables, e.g. for internal waves in a rotating stratified fluid k = (m, n, l). This can be done using the q-class decomposition. Explicit construction of q-classes for (2.54) were presented in Section 2.2, e.g. if √ √ √ √ 4 t1 + 4 t2 = 4 t3 + 4 t4 (2.67) with ti ∈ N, ti > 0 keeps true, only two cases are possible: Case 1: all the vectors belong to the same q-class Clq , so that √ √ √ √ γ1 4 q + γ 2 4 q = γ3 4 q + γ4 4 q

(2.68)

with γ1 , γ2 , γ3 , γ4 ∈ N and q a class index; Case 2: the vectors belong to two different q-classes, Clq1 , Clq2 , and √ √ √ √ γ1 4 q 1 + γ 2 4 q 2 = γ1 4 q 1 + γ 2 4 q 2

(2.69)

with γ1 , γ2 ∈ N and q1 , q2 being class indexes. In this section, we concentrate on Case 1 and in our presentation closely follow [117]. Before describing the algorithm, we need to introduce notions of the weight of the wavevector and q-class multiplicity. Definition 8. Integers γi are called weights of the wavevectors ki , i = 1, . . . , 4. Equation γ1 + γ2 = γ3 + γ4 is called the weight equation. A number γ max (q, D) is called the class multiplicity and denoted M(q).

2.3 q-class decomposition

47

For the domain D = 103 , class multiplicities are reasonably small numbers, γ max (1) = 37 being the largest. Class multiplicities for the majority of classes (starting with q = 125 002) are equal to 1, which are empty – this fact allows us to achieve a major shortcut in computation time. Step 1: Calculating relevant class indexes Every q must have a representation as a sum of two squares of integer numbers. According to the Euler’s theorem, this means that, in the prime factorization of q, every prime factor p = 4u + 3 occurs to an even degree. As γi4 evidently contains every prime factor to an even degree, this condition must also hold for q. This means that if q is divisible by a prime p = 4u + 3, it should be divisible by its square and should not be divisible by its cube. Now the computation of all q can be accomplished with a sieve-type procedure. 1. Create an array Arq = [1, . . . , 2D 2 ] of binary numbers, setting all the elements of the √ 4 array to 1. Make the first pass: for all primes p in the domain 2 ≤ p ≤ 2D 2 , set to 0 the elements of the array p4 , 2p4 , . . . , κp4 , (2.70) where κ = 2D 2 /p4 . In the second pass, for all primes p4u+3 ≡ 3(mod 4), p ≤ 2D 2

(2.71)

and integer factors a = 1, . . . , amax such that ap ≤ 2D 2 , do the following. If a = 0(mod p), then set the apth element of the array Arq to 0. If a ≡ 0(mod p), and if a ≡ 0(mod p2 ), then also set the apth element of the array Arq to 1. Notice that in the second pass, the first check should only be done for primes √ √ 3 p ≤ 2D 2 and the second check for p ≤ 2D 2 . 2. Create an array Wq of “work indexes.” In the third pass, we fill it with indexes of the array Arq for which the elements’ values have not been set to 0 in the first two passes. 3. Create an array of class multiplicities Mq and fill it with corresponding class multiplicities. All numbers q found at present do have a representation as a sum of two squares, but not all of them with |m| ≤ D and |n| ≤ D (these q will be discarded automatically at further steps). In the domain D = 103 , the number of class indices πcl (103 ) = 384 145.

Step 2: Finding decompositions into the sum of two squares In [117], decomposition of an integer into the sum of two squares has been found using the procedure suggested in [13] and Euler’s theorem. This can also be done without decomposition of an integer into a sum of two squares (Attila Pethoe, 2007, private communication). Indeed, Euler’s theorem implies that the primes to be found at this step should be divided into two classes, according to whether the prime has reminder one or three modulo 4. Accordingly, allowed qs have the form

48

Kinematics: Wavenumbers

q=

%

4|p−1

p αp ·

%

p βp ,

(2.72)

4|p−3

where αp ∈ {1, 2, 3} and βp ∈ {2}. If 4|p − 1, then there are uniquely defined positive integers ap , bp such that p = ap2 + bp2 . Being rewritten in a complex form, p = (ap + ibp )(ap − ibp ) and   % %  α q= p βp . (2.73) ap + ibpp (ap − ibp )αp · 4|p−1

4|p−3

Now let 0 ≤ γp , δp ≤ αp be integers, then the complex numbers   % %  γ q= p βp . ap + ibpp (ap − ibp )δp · 4|p−1

(2.74)

4|p−3

are all Gaussian integers with norm q. Thus precomputing the values αp and βp gives allowed indexes q at once, without finding decompositions into the sum of two squares. Step 3: Solving the weight equation Consider now the weight equation for Case 1: γ1 + γ2 = γ3 + γ4

(2.75)

with 1 ≤ γi ≤ M(q) (see 2.68). For convenience, we change our notation to γ1L , γ2L (left) and γ1R , γ2R (right) and introduce weight sum Sγ = γ1∗ + γ2∗ . Without loss of generality, we can suppose γ1L ≤ γ1R ≤ γ2R ≤ γ2L .

(2.76)

Notice that we may not assume strict inequalities because even for γi = γj there may exist two distinct vectors (mi , ni ), (mj , nj ) with m2i + n2i = m2j + n2j = γ 4 q either due to the possibility of representing γ 4 q as the sum of two squares in multiple ways or even for a single two-square representation or due to the possibility of taking different sign combinations (±|m|, ±|n|) left and right. Now we may encounter the following four situations demonstrated in Fig. 2.1. • γ1L < γ1R < γ2R < γ2L , • γ1L = γ1R < γ2R = γ2L , • γ1L < γ1R = γ2R < γ2L , • γ1L = γ1R = γ2R = γ2L .

The search is organized as follows. Each admissible sum of weights Sγ ; 2 ≤ Sγ ≤ M(q) is partitioned into the sum of two numbers

2.3 q-class decomposition

49

Fig. 2.1 Weight diagram

Sγ = γ1L + γ2L ,

1 ≤ γ1L ≤ γ2L ≤ M(q).

(2.77)

Then the same number is partitioned into the sum of γ1R , γ2R ,

γ1L ≤ γ1R ≤ γ2R ≤ γ2L .

(2.78)

Evidently, if Sγ ≤ M(q) + 1, then the minimal γ1L is 1, otherwise it is Sγ − M(q) (to provide γ1R ≤ M(q)). The maximal γ1L is always Sγ /2 and similarly γ1R ≤ Sγ /2. Interesting enough, the results of numerical simulations show that most classes have small multiplicities and then degenerate cases prevail. The numbers of classes are 24 368, 57 666, 13 987, and 63 778 for the cases 1, 2, 3, and 4 correspondingly. Step 4: Discarding “lean” classes The domain D ≤ 103 contains 384 145 q-classes, among them 357 183 q-classes have multiplicity 1, i.e. about 93% of all classes are empty and only 29 272 must be checked for probable solutions.

50

Kinematics: Wavenumbers

Step 5: Checking linear conditions: symmetries and signs This step is almost trivial. However, a few points should not be overlooked in order to organize a correct, exhaustive, and efficient search: (a) In the wholly nondegenerate case (Fig. 2.1, upper left), all four waves must be pairwise unequal. For the first degenerate case of the weight diagram (Fig. 2.1, upper right), we must provide m1L = m1R and m2L = m2R . For the second one (Fig. 2.1, lower left), m1R = m2R and for total degeneration (Fig. 2.1, lower right) m1L = m2L = m1R = m2R .

(2.79)

(b) Solutions that differ only in the transposition of two wavevectors, e.g. (0, −9)(0, 49)⇒(15, 20)(−15, 20) (0, −9)(0, 49)⇒(−15, 20)(15, 20)

(2.80)

should of course, be regarded as the same solution. (c) The set of solutions possesses some evident symmetries: if (m1L , n1L )(m2L , m2L ) ⇒ (m1R , n1R )(m2R , m2R ),

(2.81)

(−m1L , n1L )(−m2L, n2L ) ⇒ (−m1R , n1R )(−m2R , n2R ),

(2.82)

(m1L , −n1L )(m2L , −n2L ) ⇒ (m1R , −n1R )(m2R , −n2R ),

(2.83)

(−m1L, −n1L )(−m2L , −n2L ) ⇒ (−m1R , −n1R )(−m2R , −n2R ).

(2.84)

then, of course,

Taking into account these points, an effective search is constructed easily. Example 12: Ocean planetary waves To demonstrate the flexibility of our algorithm, let us take another example: threewave resonances of ocean planetary waves. In this case, resonance conditions have the form ⎧ ⎨" 1 +" 1 =" 1 , m21 +n21 m22 +n22 m23 +n23 (2.85) ⎩ m1 + m2 = m3 , and construction of q-classes is given in Section 2.2. Steps 1 and 2 of the algorithm are the same, with the set of q-class indexes being a subset of class indexes of the previous section. The weight equation is in this case 1 1 1 + = γ1 γ2 γ3

or

γ3 =

γ1 γ2 , γ1 + γ2

(2.86)

2.3 q-class decomposition

51

which has relatively few solutions in integers. Indeed, even q-class Cl1 with multiplicity 1414 contains only 3945 solutions. In this case, it makes sense to generate and store the set of triples (γ1 , γ2 , γ3 ), which constitute integer solutions to (2.86) for 1 ≤ γi ≤ M(1), and for each q to just take its subset 1 ≤ γi ≤ M(q). Discarding “lean” q-classes becomes trivial: disregarding q-classes with multiplicity 1, only 63 828 classes are left, from 243 143 in the computation domain. A check of linear conditions is also much easier than in the previous example for we have only one equation with three variables instead of two equations with four variables each. Example 13: Sound waves Our next example, three-wave resonances among sound waves, becomes trivial because resonance conditions take the form ⎧" " " ⎪ 2 2 2 2 2 2 ⎪ ⎪ ⎨ m1 + n1 + m2 + n2 = m3 + n3 , m1 + m2 = m3 , ⎪ ⎪ ⎪ ⎩n + n = n , 1

2

(2.87)

3

and there are only two minor differences from the previous example: (1) the weight equation to solve is now γ1 + γ2 = γ3 , and (2) there are two linear conditions to be checked, not one.

Two q-classes Straightforward application of classes also does not bring much in the case of two q-classes. The classes are interlocked through linear conditions and even if for each couple of classes we could detect interlocking and find solutions, if any, in O(1) operations, the overall computational complexity is at least πcl (D)2 ∼ O(D 4 ). For D = 103 , this implies 1.5 · 1011 pairwise class matches, which is outside any reasonable computational complexity limits. The trouble with this approach – as, for that matter, with virtually any algorithm consuming much more computation time than the volume of its input and output data implies – is that we perform a lot of intermediary calculations, the results of which are later discarded. Below we describe briefly the algorithm first developed in [118]. It allows us to perform every calculation with a given item of input data just once (or a constant number of times).

52

Kinematics: Wavenumbers

Let us rewrite (2.69) as ⎧



2 + n2 1/4 = m2 + n2 1/4 = γ √ 4 ⎪ m 1 q1 , ⎪ 1L 1L 1R 1R ⎪ ⎪ ⎪

1/4 2

1/4 √ ⎪ ⎪ ⎨ m22L + n22L = m2R + n22R = γ2 4 q 2 , ⎪ m1L − m1R = −m2L + m2R , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩n1L − n1R = −n2L + n2R ,

(2.88)

where q1 , q2 are two different q-class indexes and γ1 , γ2 the corresponding weights, and introduce some necessary definition. Definition 9. For any two decompositions of a number γ14 q into a sum of two squares, the value δm = mL − mR is called m-deficiency, δn = nL − nR is called n-deficiency, and δm,n = (δm , δn ) the deficiency point. Two wavevectors belonging to the same solution have obviously equal deficiencies: δ1m = m1L − m1R = −m2L + m2R = δ2m ,

(2.89)

δ1n = n1L − n1R = −n2L + n2R = δ2n .

(2.90)

For a given weight γ , every two decompositions of γ 4 q into a sum of two squares yield, in general, four deficiency points with δm , δn ≥ 0. Consider unsigned decompositions mL , mR , nL , nR ≥ 0. Assuming mL ≥ mR , nL ≤ nR , the four points are (mL + mR , nL + nR ), (mL + mR , −nL + nR ),

(2.91)

(mL − mR , nL − nR ), (mL − mR , −nL + nR )

(2.92)

and there are four (symmetrical) points in each of the other three quadrants of the (m, n) plane. γ

Definition 10. The set of all deficiency points of a q-class for a given weight, q , is called its γ -deficiency set. The set of all deficiency points of a class, q , is called its deficiency set. Example 14: Deficiency set for Cl50 The deficiency set of the q-class Cl50 is shown in Fig. 2.2 for γ = 1. As 50 = 12 + 72 = 52 + 52 = 72 + 12 , we have three decompositions into a sum of two squares, and the nonnegative deficiency points of decomposition pairs (5, 5; 7, 1), (5, 5; 1, 7), (1, 7; 7, 1)

(2.93)

2.3 q-class decomposition

53

Fig. 2.2 Deficiency set of the q-class Cl50 with γ = 1 in the positive (m, n)quadrant

are marked with diamonds, squares, and circles, respectively. They constitute a subset of 150 , consisting of 12 points with m ≥ 0, n ≥ 0. The points symmetrical with respect to the coordinate axes (altogether 36) are not shown. The crucial idea behind the algorithm is based on the following Lemma. Lemma 5. Wavevectors of two q-classes Clq1 , Clq2 belong to the same solution to (2.69) iff their deficiency sets have a nonvoid intersection, q1 ∩ q2 = ∅. Corollary 2. If the deficiency set of a q-class does not intersect with the deficiency sets of other classes, wavevectors of this class do not participate in any of the resonances described above. The sieve-like procedure for computing relevant class indexes q, admissible weights γ , etc. is the same as for the case of one q-class, although some substantial improvements can also be made at these steps [118]. The novelty of the algorithm below is in the use of the deficiency sets of the q-classes. Thus, the starting point is to declare a two-dimensional array arDef iciency(0, . . . , 2D; 0, . . . , 2D) of type byte, which serves to store and process deficiency sets of the classes. The array is initialized with all zeroes. Pass 1: Marking deficiency points In the first pass for every class q in the main domain D, deficiency set Dq is generated. After generating the deficiency set of the class for each weight γ and uniting them, the possible doubles should be discarded. Next, for every deficiency point (δm , δn ) of the q-class, we increment the value of the corresponding element of the array by 1, except elements with value 255 when the value is not changed.

54

Kinematics: Wavenumbers

Pass 2: Discarding noninteracting classes In the second pass, we generate deficiency sets once more, and for every point of the deficiency set of a q-class we check the values of the corresponding point of arDeficiency. The values that are equal to 1 are discarded. For the problem considered, this pass excludes just 313 classes, so the time gain is very modest. However, we include this step in the presentation for two reasons. First of all, it had to be done as there was no possibility of reducing the number of classes considered, as much as possible and as soon as possible, before the most timeconsuming steps may be neglected. Second, although not much is gained for the solution to the problem at hand, the elimination techniques may play a major role in further applications of our algorithm. Pass 3: Linking interaction points to interacting vectors In the third pass, for every deficiency point δm,n and q-classes not discarded in the previous pass, values of q, γ , mL , nL , mR , nR are stored. Then the corresponding points of arDeficiency and each point where the value is larger than 1 are linked to (q, γ , mL , nL , mR , nR ). Pass 4: Gathering interaction points In the fourth pass, we go through the array arDef iciency once more and store every point with value greater than one in an array arDef iciencySol(1, . . . ; 0, . . . , 1). We also relink structures linked to deficiency points to corresponding points of the new array. Pass 5: Extracting solutions  At this pass, an array of points

δm,n and to each of these points ia list of iwe have i i i i i structures q , γ , mL , nL , mR , nR with no less than two different q . Every combination of two structures linked to the same point and having different q i yields a solution to (2.65). From every solution found, symmetrical solutions can be found by changing the signs of mi , ni . More details on the implementation (language-specific shortcuts, data structures, estimates of computational complexity, etc.) can be found in [118]. Implementation results Some overall numerical data presented in [117, 118] are given in Figs 2.3 and 2.4. The number of solutions for the two-class case, depending on the partial domain, is shown in the Fig. 2.3. Both curves are almost ideal cubic lines. Very probably they are cubic lines asymptotically. Partial domains chosen in Fig. 2.3 are of two

2.3 q-class decomposition

55

6,E+08 5,E+08 4,E+08 3,E+08 2,E+08 1,E+08 0,E+00 0

100

200

300

400

500

600

700

800

900

1000

Fig. 2.3 Number of solutions in partial domains mi , ni ≤ D and m2i + n2i ≤ D 2 , shown by curves marked squares and circles respectively

Fig. 2.4 In the circle of width 50, only solutions drawn with solid lines are taken into account, not solutions drawn with dashed line

types: squares mi , ni ≤ D, just for simplicity of computations, and circles m2i + n2i ≤ D 2 , a more reasonable choice from the physical point of view (in each circle all wavelengths are ≤ D). The curves in Fig. 2.3 are very close to each other in the domain D ≤ 500, although the number of integer nodes in a corresponding square is D 2 and in a circle with radius D there are only π D 2 /4 integer nodes. This indicates a very interesting physical phenomenon: most of the solutions have wavevectors parallel and close to either axis X or axis Y. On the other hand, the number of solutions in rings (D − 50)2 < m2i + n2i ≤ D 2 (corresponding to wavelengths between D − 50 and D) grows linearly almost perfectly. Of course, the number of solutions in a circle is not equal to the sum

56

Kinematics: Wavenumbers 105

5 × 104

0

0

10

20

30

40

50

60

70

80

Fig. 2.5 The multiplicities histogram. Vertical and horizontal axes denote the number of wavevectors and their multiplicity, respectively

of solutions in its rings: a solution lies in some ring iff all its four vectors lie in that same ring (see Fig. 2.4). That is, by studying solutions in the rings only, we exclude automatically a lot of solutions containing vectors with substantially different wavelengths simultaneously. This “cut” of sets of solutions can be of use when interpreting the results of laboratory experiments performed in investigations of waves with some given wavelengths (or frequencies) only. Another important characteristic of the structure of the solution set is wavevector multiplicity, which describes in how many solutions a given vector participates. The multiplicity histogram is shown in Fig. 2.5. On the axis X, the multiplicity of a vector is shown, and, on the axis Y , the number of vectors with a given multiplicity. This histogram has been cut off; multiplicities go very high – indeed the vector (1000,1000) takes part in 11 075 solutions. Similar histograms computed for different one-class cases show that most wavevectors, usually 70–90% for different types of waves, take part in one solution, e.g. they have multiplicity 1. The number of vectors with larger multiplicities decreases exponentially when the multiplicity is growing. The very interesting fact in the two-class case is the existence of some initial interval of small multiplicities, from 1 to 10, with a very small number of corresponding vectors. For instance, there exist only seven vectors with multiplicity 2. Beginning with multiplicity 11, the histogram is similar to that in the one-class case. This form of the histogram is quite unexpected and demonstrates once more the specifics of the two-class case compared with the one-class case. As we can see from the multiplicity diagram, the major part in the two-class case is played by much larger groups of waves with the number of elements being of order 40: each solution consists of four vectors, groups contain at least one vector with multiplicity 11, although some of them can take part in the same solution. We will study these groups in more detail in Chapter 3.

2.4 Rational dispersion function

57

2.4 Rational dispersion function An example of a rational dispersion function, given by atmospheric planetary waves, has the form ω ∼ m/[n(n + 1)] and has been studied first in [119]. Correspondingly, three-wave resonance conditions read ⎧ ⎪ ⎪ ω1 + ω2 = ω3 , ⎪ ⎪ ⎪ ⎪ ⎪ m1 + m2 = m3 , ⎪ ⎪ ⎪ ⎪ ⎨|n1 − n2 | ≤ n3 ≤ n1 + n2 , ⎪ ⎪ n1 + n2 + n3 is odd, ⎪ ⎪ ⎪ ⎪ ⎪ mi ≤ ni ∀i = 1, 2, 3, ⎪ ⎪ ⎪ ⎪ ⎩ ni = nj ∀i = j,

(2.94)

with dispersion function ω ∼ m/n(n + 1). It is known that (2.94) has an infinite number of integer solutions [126]. Indeed, one-parametric series ! n1 = 6p, n2 = 6p + 2, n3 = 6p + 1, m1 = 2p, m2 = 2p + 1, m3 = 4p + 1

(2.95)

satisfies (2.94) for any integer p. This can easily be checked by direct substitution. A more complicated example of the two-parametric series of solutions has the form ⎧ ⎪ n = (c2 + c)s + c, n2 = cs, n3 = (c + 1)s, ⎪ ⎨ 1 (2.96) m1 = (2cs + s + 1)/f, m2 = (c2 s + c − s)/f, ⎪ ⎪ ⎩ m3 = (c2 s + 3cs + c + 1)f. √ Here c is an arbitrary irreducible fraction p/q in the interval (1, (1 + 5)/2), and s = A(p, q)t + B(p, q), f = f (p, q), while A(p, q),

f (p, q),

B(p, q)

(2.97)

are certain integer-valued functions of the arguments p, q and t ∈ N. The algorithm used to obtain suitable functions A(p, q), f (p, q), B(p, q) can be found in [110]. Nevertheless, these are not all the solutions to this system, and in a given spectral domain we have to compute them numerically. In the previous sections, algorithms based on q-class decomposition were presented for finding nonlinear resonances for irrational dispersion functions. The

58

Kinematics: Wavenumbers

key points of the presentation were, first, that these algorithms differ only in some details and their core is applicable to a wide class of dispersion functions, thus justifying the name of “generic.” Second, irrational equations in integers were solved without use of floating-point arithmetic and without even resolving the irrationalities involved. This gave us an enormous gain both in performance time and orders of numbers used. A rational dispersion function demands some additional consideration, while q-class decomposition is not applicable in this case and a special algorithm is needed. The straightforward approach to solving (2.94) would be to get rid of denominators ni (ni + 1), substitute m3 with m1 + m2 , and perform a full search on m1 , m2 , n1 , n2 , n3 . This implies D 5 computation time and operating with numbers of the order of D 5 , computation time and order of numbers used growing rapidly with the domain size. We are going to present a far more efficient algorithm based on a simple observation. Any equation in rational functions in integers can be trivially transformed into a Diophantine equation: ⎛ ⎞  Pi  % ⎝P i =0 ⇒ Qj ⎠ = 0. (2.98) Qi i

i

j

The idea underlying the algorithm, first presented in [119], can be illustrated by a simple consideration. Before solving in integers an equation a=b

P , Q

0 < a ≤ a0 ,

0 < b ≤ b0

(2.99)

P is an integer with P /Q being an irreducible fraction, notice that the number b Q only if b is a multiple of the denominator Q. Then (a, b) is a solution only if b = kQ with integer k, i.e. a = kP and (kP , kQ) is a solution for any

k, 1 ≤ k ≤ min(P /a0 , Q/b0 )

(2.100)

and the solutions of the equation are exhausted. Notice that there is no search at all involved. Step 1: Search on n1 , n2 , n3 The search on n1 , n2 , n3 is organized conventionally. Without loss of generality, consider n1 < n2 . Notice that n3 always lies between n1 and n2 and from the two “triangle inequalities” of (2.94), the second one always holds, while the first one implies n3 > n2 − n1 , which limits the search on n3 if n2 − n1 > n1 . The oddity condition allows us to run the cycle on n3 in steps of 2.

2.4 Rational dispersion function

59

Up to now, the computational complexity is O(D 3 ). Step 2: Cycle elimination on m1 , m2 Let us introduce notation bi = ni (ni + 1) and rewrite the first equation of (2.94) as m1 /b1 + m2 /b2 = m1 /b3 + m2 /b3

(2.101)

or

b3 − b1 b2 · . (2.102) b2 − b3 b1 After reducing the numerator and denominator of the fraction on the right side by their greatest common divisor GCD, (2.102) reads m2 = m1 ·

m2 = m1 ·

RN RD

(2.103)

and every solution has the form m1 = kRD , m2 = kRN ,

k ≤ min(n1 /RD , n3 /(RN + RD )).

(2.104)

The second condition follows from m3 = m1 + m2 ≤ n3

(2.105)

and is stronger than m2 ≤ n2 . The computational complexity of the whole algorithm is thus O(D 3 log D), D 3 for the cycle on ni and log D for the GCD. Remark 2.1. The algorithm above implies operating with numbers of the order of D 4 in one certain place, namely transforming RN b3 − b 1 b2 · ⇒ . b2 − b3 b1 RD

(2.106)

This could lead to overflows with D large and the computer small, say D = 1000 and a 32 bit computer or D = 106 and a 64 bit computer. There is, however, an elegant way to avoid difficulties at this point, which we describe in the next step. Step 3: Avoiding multiplications Given the fraction product, we first reduce b3 − b1 and b2 − b3 by their GCD, then b2 and b1 by their GCD. This leaves us with a product of two irreducible fractions (r31 /r23 ) · (r2 /r1 ). Now we reduce crosswise: r31 and r1 , r23 and r2 . The last reduction gives an “irreducible product” of two fractions (rr31 /rr23 ) · (rr2 /rr1 ), i.e. had we performed the multiplications, the resulting fraction would stay irreducible. The reduction schema is presented in Fig. 2.6.

60

Kinematics: Wavenumbers

b3

b1

GCD

b2

b2 GCD

b3

b1

r31 GCD

r2

rr31

rr2

rr23

rr1

GCD

r23

r1

Fig. 2.6 Bringing a product of two fractions to complete irreducibility without multiplying

We still do not perform multiplications for fear of an overflow. But now it is evident that a solution can only exist if rr23 ≤ n1 , rr1 ≤ n1 , rr31 ≤ n2 , rr2 ≤ n2 . We first check these inequalities; if one or more of them do not hold, we proceed with the n-cycle, otherwise we may safely perform multiplications (both products do not exceed D 2 ) and look for solutions. More implementation details can be found in [119], the results can be briefly summarized as follows: altogether 7282 solutions of equation (2.94) have been found in the domain D = 103 , the number of solutions grows almost linearly with enlarging the spectral domain, more than 90% of all wavevectors take part only in one solution, and wavevector multiplicity decreases exponentially with the number of solutions. All algorithms presented in this section have been programmed in programming language C++. Some results on their implementation in Mathematica, where we made use of some in-built procedures, can be found in [127]. 2.5 General form of dispersion function Let us now consider the dispersion relation in the form ω = ω(k),

k = (m, n),

m, n ∈ Z,

(2.107)

where Z denotes the set of integers, i.e. the dispersion function is an arbitrary realvalued function of integer variables. Without loss of generality, we can assume that k is a two-dimensional vector. Rewrite (1.74) as ω(m1 , n1 ) ± ω(m2 , n2 ) ± . . . ± ω(ms , ns ) = 0,

S ∈ Z,

(2.108)

and regard ω as a real-valued function defined on the Cartesian product, ω : Z × Z → R, and let  = I m(ω) be its image. It is evident that the image is countable (as an image of a countable set),

2.5 General form of dispersion function

 = {v1 , v2 , . . . , vp , . . .}.

61

(2.109)

Let V1 , . . . , Vp be subsets of R, defined as follows. Each Vi is composed of all rational numbers multiplied by some vi ∈ . Therefore, the Vi may be regarded as a one-dimensional vector space over the field of rational numbers Q. It is evi* dent that 0 ∈ Vi ∩ Vj , ∀i, j and, therefore, Vi Vj = ∅ . Since all the Vi have dimension 1, the intersection Vi ∩ Vj can be nonzero only in the case Vi = Vj . Notice that the indexing of the vector spaces Vi by representation (2.109) leads to enumeration with repetitions: indeed, if vi = rvj , where r ∈ Q, then the corresponding vector spaces Vi and Vj coincide (as sets). Later on we shall consider only pairwise unequal spaces Vi = Vj . In that case, the indexing should have been, strictly speaking, changed; we do not do this, allowing ourselves the small liberty of notations. Now let ω˜ : Z × Z ⇒ {Vi } i = 1, 2, . . . , p . . .

(2.110)

be the function mapping the vector (m, n) to the vector space generated by the image of ω(m, n) over Q. Theorem 8 (Theorem on partitioning). Let the set of positive integers be partitioned as N = I 1 ∪ I 2 . . . ∪ Il . . . ,

Ii ∩ Ij = ∅, i = j, Il = i1l , i2l , . . . , itl

so that the set of vector spaces + = Vi 1 ⊕ Vi 1 ⊕ · · · ⊕ Vi 1 ⊕ · · · VI1 = i∈I1

... VIl =

+ i∈Il

1

2

k

= Vi l ⊕ Vi l ⊕ · · · ⊕ Vi l ⊕ · · · 1

2

k

(2.111)

(2.112)

(2.113)

... possesses the property of “global linear independence,” i.e. vi1 + vi2 + · · · + vis = 0,

vij ∈ VIj

(2.114)

if not all the vIj are equal to 0 and s is the integer that appears in (2.108). Then the set of vectors that are components of solutions of (2.108) can be partitioned into disjoint classes Cl1 , . . . , Cll . . . each of which lies in a unique set from

62

Kinematics: Wavenumbers



 Cl1 ⊆ ω˜ −1 (VI1 ) = ω˜ −1 Vi 1 ∪ ω˜ −1 Vi 1 ∪ · · · 1

1

(2.115)

...



 Cll ⊆ ω˜ −1 (VIl ) = ω˜ −1 Vi l ∪ ω˜ −1 Vi l ∪ · · · , 1

1

(2.116)

where by ω˜ −1 (V ) we mean pre-image. Corollary 3. In the notation of the theorem, let all the Il be one-element sets: I1 = {i1 },

I2 = {i2 }, . . . ,

Il = {il }

and V1 , V2 , . . . , Vl possess the global linear independence property. Then Cl1 ⊆ ω˜ −1 (V1 ), . . . , Cll ⊆ ω˜ −1 (Vl ).

(2.117)

(2.118)

The proof of this theorem can be found in [110], while the corollary is easily derived from the theorem by direct renumbering. This theorem is not constructive in the sense that it neither gives an algorithm for constructing such a partition into classes when it exists, nor even yields necessary and sufficient conditions for the existence of such a partition. It only enables us to construct the partition of the set of wavevectors satisfying (2.108) when the partition of vector spaces of a special form is already given. The problem of the existence of the latter must in each case be solved as such and as a rule does lead to the study of linear independence of sets of algebraic or transcendental numbers. Actually, all examples above are applications of not the theorem but of Corollary 3 (in fact, they gave rise to the statement of the corollary). Example 15: Ocean planetary waves Coming back to the example of ocean planetary waves studied in Section 2.2, we have (in the notation of Theorem 8) ,p√ = q, p, s, q ∈ N, q is square free , (2.119) s √

√

˜ i , ni ) ∈  q i (2.120) Vi = Q qi , ω(m and the condition of global independence of {Vi } is a corollary of Lemma 3. * It follows that Vi Vj = {0} and the conditions of Theorem 8 are satisfied, i.e ω˜ −1 (V1 ), . . . , ω˜ −1 (Vj ) are disjoint sets and Cl1 ∈ ω˜ −1 (V1 ), . . . , Clj ∈ ω˜ −1 (Vj ), . . .

(2.121)

2.5 General form of dispersion function

63

Remark 2.2. In this chapter, we have regarded various examples of rational and irrational dispersion functions. Transcendent dispersion functions are also known: for instance, ω2 = g k tanh k d, which is the dispersion function for gravity surface waves in water, g and d being the gravitational constant of acceleration and average water depth, respectively. To apply Theorem 8, we refer first to the Lindemann–Weierstrass theorem [9]: Theorem 9 (Lindemann–Weierstrass). If α1 , . . . , αn are algebraic numbers that are linearly independent over the field of rational numbers Q, then eα1 , . . . , eα1 are algebraically independent over Q. Now, the standard representation tanh x =

e2x − 1 e2x + 1

(2.122)

allows us to re-write (k1 tanh k1 d)1/2 + (k2 tanh k2 d)1/2 = (k3 tanh k3 d)1/2 + (k4 tanh k4 d)1/2

(2.123)

as a linear√combination of exponents of different algebraic numbers (remember that k = m2 + n2 ) with the coefficients depending on ki ). Combination of the Weierstrass–Lindemann theorem and q-class decomposition will produce partitioning according to Theorem 8. A sketch of the computation is given in [104], Discussion. Remark 2.3. In this book, we present a detailed study of nonlinear resonances of two-dimensional waves, i.e. with wavevectors of the form k = (m, n). However, as was shown in the example of internal waves, (2.33), nonlinear resonance analysis can be performed along the same lines as for the case of three-dimensional waves. The only computational difference would be to compute the decomposition of k = |k| into the sum of three squares, not two squares.

3 Kinematics: Resonance clusters

Ivan worked assiduously, crossing out what he had written, putting in new words, and even attempted to draw Pontius Pilate and then a cat standing on its hind legs. M. Bulgakov Master and Margarita

3.1 An easy start The classical way to represent nonlinear resonances graphically is by the construction of so-called resonance curves. A resonance curve is the locus of pairs of wavevectors interacting resonantly with a given wavevector. Its construction, very popular in the 1960s, gives some useful, though partial, information about possible resonances in a wave system. For instance, in [152] and [195] resonance curves are regarded for three- and four-wave resonance systems, respectively. The form of the resonant curve changes drastically according to the magnitudes of the wavevectors. A locus might be an ellipse, be shaped like an hour-glass, or even degenerate into a couple of lines. Examples of resonance curves for ocean planetary waves, with a dispersion function of the form (2.21), are shown in Fig. 3.1. Without knowledge of the wave numbers corresponding to the complete resonance set, all this information is only qualitative, of course. Eventually, the characteristic form of loci is useful to know before planning any laboratory experiments. For instance, in [195] a special type of resonant quartet of surface gravity waves is studied in which two wavevectors coincide. This special configuration was chosen because it “is convenient for experimental study” [195].

64

3.1 An easy start

65

Fig. 3.1 Dispersion function ω ∼ (m2 + n2 )−1/2 . Left panel: m1 = 1, n1 = 1, 4m2 ∈ [−20, 20]. Right panel: m1 = 5, n1 = −7, m2 ∈ [−20, 20]

Resonance curve representation has two drawbacks: • First, it allows us to visualize only a part, if any, of the resonance set, namely all

wavevectors interacting resonantly with a given wavevector which is fixed. In the case of four-wave resonances, we have to fix two wavevectors to construct resonance curves. If the complete resonance set is not known, it can happen that the constructed resonance curve will not contain any integer points. An example can be given by the resonance curve constructed for wavevector (1, 3) in the case of ocean planetary waves, ω ∼ (m2 + n2 )−1/2 , which is not a part of any solution. • Second, there is no constructive way to establish whether there exist any integer points on these curves [165]. As we have seen in Section 2.2, it can happen that the equation ω1 + ω2 = ω3 has no integer solutions for any wavevector.

The graphical presentation of the complete resonance set for the case of twodimensional wavevectors, first suggested in [110] for three-wave resonances, is rather general and provides a simple way to exhibit a complete resonance set. The construction is performed in two steps. First, we regard each 2D-vector with integer coordinates as a node of the integer lattice and connect these nodes, which construct one solution (triad, quartet, etc.) The result is called the geometrical structure of the resonance set. Second, we single out all “similar” elements of the geometrical structure, i.e. all connected graph components of the same form, and compute the number of elements for all different forms. Definition 11. Any connected graph component of the geometrical structure of the resonance solution set is called the resonance cluster. In the three-wave system, examples of possible clusters are given by isolated triangles, two triangles connected via one node, two triangles connected via two nodes,

66

Kinematics: Resonance clusters

Fig. 3.2 Left panel: Example of geometrical structure, spectral domain |ki | ≤ 50 and q ≤ 300. Right panel: Topological structure, the same spectral domain. The number in brackets shows how many times a corresponding cluster appears in the chosen spectral domain

three triangles connected in a few different ways, etc. As a result, we get a list of clusters and a number attributed to each element of the list. This list is called the topological structure of the resonance set. Examples of geometrical and topological structures are shown in Fig. 3.2, for the complete resonance set in the case of ocean planetary waves, with dispersion function ω ∼ (m2 + n2 )−1/2 . Obviously, the geometrical structure, shown in the left panel, is too nebulous to be useful, even in relatively small spectral domains. On the other hand, the topological structure, shown in the right panel, is quite clear and gives us general information about the dynamical system covering the behavior of each resonance cluster. Remembering the form of the primary dynamical system (1.129) with coupling coefficient Z, we can see immediately that the form of the dynamical system corresponding to a generic cluster might depend on the connecting modes within a cluster. Indeed, equations for some modes have coefficients Z on the left side of (1.129), and for some others have coefficients −Z. This fact can be included into the topological representation of the resonance solution set, in the form of marked vertexes and/or arcs. The result is called NR-diagram; its form allows us to reconstruct a unique dynamical system for a cluster. In Chapter 3, we learn how to construct NR-diagrams for resonance clusters in three- and four-wave systems, and restore the corresponding dynamical system. 3.2 Topological structure vs dynamical system Let us begin with dynamical equations for a triad: B˙ 1 = Z1 B2∗ B3 ,

B˙ 2 = Z1 B1∗ B3 ,

B˙ 3 = −Z1 B1 B2 .

(3.1)

3.2 Topological structure vs dynamical system

67

To write out the dynamical equations for a resonance cluster consisting of two triads connected via one mode, we have to take two copies of equations (3.1), ! B˙ 1 = Z1 B2∗ B3 , B˙ 2 = Z1 B1∗ B3 , B˙ 3 = −Z1 B1 B2 , (3.2) B˙ 4 = Z2 B ∗ B6 , B˙ 5 = Z2 B ∗ B6 , B˙ 3 = −Z2 B4 B5 . 5

4

Supposing that, say, B1 = B4 yields ⎧ ⎪ B˙ = Z1 B2∗ B3 + Z2 B5∗ B6 , ⎪ ⎨ 1 B˙ 2 = Z1 B1∗ B3 , B˙ 3 = −Z1 B1 B2 , ⎪ ⎪ ⎩˙ B5 = Z2 B ∗ B6 , B˙ 6 = −Z2 B1 B5 .

(3.3)

1

Proceeding this way, we can write out dynamical systems for the arbitrary resonance clusters shown in Fig. 3.2. This would be an easy exercise for the first two or three types of clusters, but quite a boring task for clusters of a more complicated structure. On the other hand, this procedure can easily be programmed in any symbolical programming language. This means that knowledge of the complete resonance set and its topological structure can automatically be transformed into a few small independent dynamical systems (3.1), (3.3), etc. with known coupling coefficients. We can see that triangles on the (m, n) lattice, corresponding to resonant triads are, so to say, simple bricks used to construct all other resonance clusters of an arbitrary structure. Therefore, they are called primary clusters, i.e. a triad and a quartet are primary clusters in three- and four-wave systems respectively. Aiming to construct a cluster and its dynamical system, let us first give a definition of isomorphic dynamical systems. Definition 12. Dynamical systems of the form (3.1), (3.3), etc. are called isomorphic, if one can be transformed into another by a change of indexes. Otherwise, they are called nonisomorphic. Notice that (3.1) is symmetric with respect to the change of indexes 1 ⇔ 2 (respectively 4 ⇔ 5), but is not symmetric with respect to the change of indexes 1 ⇔ 3 or 2 ⇔ 3. This means that connections B1 = B5 , B2 = B4 , B2 = B5 will generate dynamical systems isomorphic to (3.3). However, connections B1 = B6 , B2 = B6 , B3 = B4 , B3 = B5 generate another set of isomorphic dynamical systems, which are not isomorphic to (3.3). For instance, connection B3 = B4 yields ⎧ ⎪ B˙ = Z1 B2∗ B3 , B˙ 2 = Z1 B1∗ B3 , ⎪ ⎨ 1 (3.4) B˙ 3 = −Z1 B1 B2 + Z2 B5∗ B6 , ⎪ ⎪ ⎩˙ B5 = Z2 B3∗ B6 , B˙ 6 = Z2 B4∗ B5 .

68

Kinematics: Resonance clusters B5

B1

B4

B5

B2

B3

B1

B6

B4

B2

B3

B6

Fig. 3.3 Example of isomorphic graphs and unisomorphic dynamical systems. The left graph corresponds to the dynamical system (3.6) and the graph on the right corresponds to the dynamical system (3.7)

Even in the case where variables Bj are real, this difference does not disappear, due to the minus sign in the equation for the mode with highest frequency ω3 . The dynamics of this mode is special, while resonance conditions have been chosen in the form ω1 + ω2 = ω3 . For clarity of presentation, we will not go into these details till the very end of this chapter. Instead, we introduce notation for (3.1) as (B1 , B2 , B3 ) with ordering by the increasing of indexes; this allows us to denote (3.3) and (3.4) as (B1 , B2 , B3 )(B1 , B5 , B6 )

and (B1 , B2 , B3 )(B3 , B5 , B6 )

(3.5)

respectively. All dynamical systems presented below are written for brevity in real form. Notice that even in this simplified setting, the topological structure described above does not grant one-to-one correspondence between a resonance cluster of a given form and a dynamical system in the generic case. Indeed, all isomorphic graphs presented in Fig. 3.2 are covered by similar dynamical systems, where only the magnitudes of the coupling coefficients vary. However, in the general case, graph structure thus defined does not represent the dynamical system unambiguously, which is illustrated by Fig. 3.3. Two objects shown there are isomorphic as graphs but represent two different dynamical systems (B1 , B2 , B3 ), (B1 , B2 , B5 ), (B1 , B3 , B4 ), (B2 , B3 , B6 )

(3.6)

(B1 , B2 , B5 ), (B1 , B3 , B4 ), (B2 , B3 , B6 ).

(3.7)

and

The first describes four connected triads, while the second, only three (a placeholder was put inside the triangle not representing a resonance solution). Below we will develop, step by step, a graphical representation providing the isomorphism of topological elements and dynamical systems. This construction was first developed in [124], for three-wave resonance systems.

3.3 Three-wave resonances

69

3.3 Three-wave resonances Definition 13. Let us first regard a plane graph Gt constructed of triangles, and the list of all triangles corresponding to the solutions of resonance conditions, Lc . The number of elements in Lc is called the order of Gt and denoted as N (Gt ). Each triangle of Gt is called a three-cycle. Each triangle that does not appear in the list Lc is called an empty three-cycle. The structure (Gt , Lc ) is called an i-pair (i for isomorphism). Consider an i-pair (Gt , Lc ) consisting of: • a graph Gt , each edge of which belongs to at least one three-cycle; and • a nonempty list Lc of some three-cycles of length 3 of Gt such that each edge of Gt

belongs to some cycle(s) of Lc .

Definition 14. The number of occurrences of each vertex v ∈ Gt in Lc is called vertex multiplicity and is denoted as μ(v). The number of different vertices in Lc is denoted as M(Gt ). Our idea is to construct a set of all possible graphs of this type of order ≤ N for some given N inductively, beginning with a single triangle. As the next step, we can choose all nonisomorphic graphs from this set and compare the corresponding lists Lc to find all different dynamical systems. (N−1) At the first induction step, Gt consists of one triangle. Let Gt be a graph constructed at the (N − 1) induction step; it consists of N1 triangles. Let the new N th triangle be T = {v1 , v2 , v3 }, then all possible ways to “glue” it to G(N−1) are t via a vertex, an edge, and a vertex and an edge simultaneously, i.e. via one to three vertices (see Figs 3.4–3.6). Vertex gluing. In this case, one, two, or three vertices of the new triangle are identified with the (glued to) vertices of some distinct triangles of the graph, con1) structed at the previous inductive step, G(N . In this case, the G(N−1) structure t t is extended by: (1) two vertices and three edges (one vertex glued), (2) one vertex and three edges (two vertices glued), and (3) three edges (three vertices glued). Edge gluing. In this case, one, two, or three edges and the corresponding vertices of the new triangle are glued to the edges and vertices of some distinct adjacent triangles of the graph. Notice that gluing of three edges is simply filling an empty (N−1) (N−1) triangle of Gt . In this case, the Gt structure is extended by: (1) one vertex and two edges (one edge glued), (2) one edge (two edges glued), and (3) the graph structure stays unchanged (three edges glued). Mixed gluing. In this case, one vertex vN1 of the new triangle is glued to a vertex of some triangle of the graph and the edge vN2 vN3 is glued to an edge of another structure is enhanced by two edges. triangle. By mixed gluing, the G(N−1) t

70

Kinematics: Resonance clusters

Fig. 3.4 (a): Gluing by one vertex. (b): Gluing by two vertices. (c): Gluing by three vertices

(a)

V1

V2

V2

V1 V3

V3

(b) V2 V1

V1 V2 V3

V3

(c) V2

V1

V1

V2 V3 V3

Fig. 3.5 (a): Gluing by one edge. (b): Gluing by two edges. (c): Gluing by three edges

3.3 Three-wave resonances

71

V1

V2 V1

V3

V3

V2

(N−1)

Fig. 3.6 Mixed gluing of a new triangle to Gt (N−1)

(N−1)

In each case, the list Lc of graph Gt is extended by the cycle vN1 vN2 vN3 of the new triangle (or whatever these vertices are called after the gluing). Enhancing a given Gt with V vertices and E edges by a triangle, we encounter the following possibilities: • a sole triangle not connected to the existing graph is added (one possibility); • vertex gluing (V possibilities); • edge gluing (E possibilities); • mixed gluing (approximately E/V (E/V − 1) possibilities); • filling an empty triangle (very rare).

Therefore, at each inductive step the mean number of vertices is V ≤ 1.5N and the number of edges can be roughly estimated as E ≤ 2N . Accordingly, the number of emerging nonisomorphic graphs can be estimated from the above as some ∼ 4N and the overall number of graphs at step N is O(N 2 ).

Hypergraph presentation To diminish computational time and complexity, we construct the hypergraph representation of an i-pair (Gt , Lc ) introduced in the previous section. A hypergraph is a structure that consists of a set of vertices and a multiset of edges, called hyperedges. A hyperedge is a set of vertices; all vertices in such a set are connected. The collection of hyperedges is a multiset because it is possible that some hyperedges appear more times than once. A traditional graph is a special case of a hypergraph in which all edges are two-element sets and do not appear more than once. For the representation of three-wave resonances, we consider the triangles as “vertices” of the corresponding hypergraph. Definition 15. A hypergraph with three-cycles of a triangle graph Gt as its vertices and vertices (m, n) of Gt as its edges is called a triangle hypergraph and is denoted as H Gt . The sets of its vertices and edges are denoted as VH G and EH G respectively, i.e. H Gt = (VH G , EH G ).

72

Kinematics: Resonance clusters

Notice that since a vertex (m, ˜ n) ˜ of Gt can belong to several three-cycles, the corresponding H Gt has in fact hyperedges instead of edges of a simple graph. A hypergraph H Gt generated by Gt has two properties: (1) Each vertex is incident to exactly three hyperedges. It follows from the fact that each vertex of H Gt represents a three-cycle, which consists of three different nodes of Gt . (2) Each pair of vertices is connected by at most two hyperedges. Indeed, two associated three-cycles of Gt have three nodes in common; hence they are identical.

Example 16: Hypergraph representation for (3.6) and (3.7) As an illustrative example, let us write out explicitly the hypergraph representation of dynamical systems (3.6) and (3.7) presented in Fig. 3.3 in the left and right panels, respectively:

VH G = {1, 2, 3, 4}, , - EH G = {2}, {3}, {4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4} (3.8) and

, - VH G = {1, 2, 3}, EH G = {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} .

(3.9)

Incidence matrix For computational purposes, it is convenient to represent a hypergraph H Gt by its incidence matrix, which is constructed in the following way. Definition 16. A rectangular matrix F = (fi,j ) with M(Gt ) columns and N (Gt ) rows is called the incidence matrix of Gt if ! 1, j th nonempty three-cycle contains ith node, fi,j = (3.10) 0 otherwise. Each column of the matrix F represents a triangle in the solution set of (2.1), while each row represents a node. Since we are not interested in the nodes themselves, but in their relation to each other, we can relabel the nodes of the triangle with ascending integers in an arbitrary way and use the labels of the nodes for indexing elements in a matrix. Now we can construct the hyperedges of H Gt : if the j th entry of a row is equal to 1, then we add j to this hyperedge. The vertices of H Gt are elements of Lc . The ordering of the hyperedges is not important, because they are a multiset. However, it is better to have a “normal form,” so we sort the

3.3 Three-wave resonances

73

hyperedges using some ordering. Since we will be interested in an implementation in Mathematica, we choose the ordering that orders lists ascending by their length, and lists of the same length lexicographically by their elements. For dynamical systems, there is no ordering that has practical advantages for implementation, so we leave them unsorted. Example 17: Incidence matrices for (3.6) and (3.7) Incidence matrices of dynamical systems (3.6) and (3.7) have the form ⎞ ⎛ ⎞ ⎛ 1 1 0 1 1 1 0 ⎜ 1 0 1 ⎟ ⎜ 1 1 0 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 1 1 ⎟ ⎜ 1 0 1 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 1 0 ⎟ and ⎜ 0 1 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ 1 0 0 ⎠ ⎝ 0 1 0 0 ⎠ 0 0 1 0 0 0 1

(3.11)

respectively. Incidence matrices of their hypergraphs that read as ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 1 1 1

1 0 0 1 1 0

0 1 0 1 0 1

0 0 1 0 1 1

⎞

⎛

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

and

1 0 0 1 1 0

0 1 0 1 0 1

0 0 1 0 1 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.12)

are also different. Incidence matrices of a dynamical system and of the corresponding hypergraph presentation are not identical but permuted. The reason is the use of different orderings for vertices. Since we use special orderings for the hyperedges, which are described by the rows of the incidence matrix, we obtain permuted rows. To identify isomorphic dynamical systems, it is not necessary to preserve ordering, because dynamical systems with permuted elements are still isomorphic. Hence, neither row permutations nor column permutations destroy the isomorphism of dynamical systems. In this example, only row permutations occur; permutations of columns are just another ordering of the elements of Lc . This construction can be reversed and the dynamical system can be reconstructed from its hypergraph representation: by considering the columns of this matrix, we know which nodes belong to a certain three-cycle. Obviously, if two hypergraphs are not isomorphic, their incidence matrices are also different. But in general for the final decision it is necessary to have an algorithm to establish the isomorphism of the hypergraphs. Since there are not many general algorithms for hypergraphs, we have to find a representation where it would

74

Kinematics: Resonance clusters T1

T1

1 2 1

2

2

1 3

3

T2

1

2

T4 3

T3

T2

3

T3

Fig. 3.7 The left multigraph corresponds to the dynamical system (3.6) and the multigraph on the right to the dynamical system (3.7)

be possible to use standard algorithms for graph isomorphism. This leads us to auxiliary multigraph construction presented in the next section. Multigraph construction A multigraph MGt is constructed in the following way. Its vertices coincide with the vertices of H Gt and each hyperedge is replaced by all two-element subsets. To preserve the complete information, we have to label the created edges so that edges that belong to the same hyperedge of H Gt are labeled identically. These labels allow us to reconstruct H Gt and F, which is a necessary step for generation of dynamical systems. Hyperedges that contain only one vertex can be omitted because they contain no further information about the cluster structure. Of course, some edges may occur in MGt twice – this is the case if two threecycles of Gt share two nodes. Fig. 3.7 shows two multigraphs corresponding to the dynamical systems shown in Fig. 3.3. For easier distinction, we use triangle symbols for the vertices of the multigraphs, because a vertex represents a threecycle of Gt . Below we list a few simple properties of multigraph MGt which are important for further algorithm implementation: • at most two edges connect a pair of vertices; • at most three differently labeled edges can meet at a vertex; • the number of vertices is equal to the number of nonempty three-cycles in Gt ; (p−1)p • the total number of edges with identical labels is , where p is the number of 2

elements in the corresponding hyperedge.

The first three properties are obvious, while the last follows from the fact that edges with identical belong to the same hyperedge, and the number of two labels

element subsets is p2 . Let us summarize briefly all the constructions described above. For a complete solution set of resonance conditions (2.1), four objects have been introduced: an

3.3 Three-wave resonances

75

i-pair (Gt , Lc ), a hypergraph H Gt , an incidence matrix F (Gt ), and a multigraph MGt . The last two objects were introduced for computational purposes, but can also be used for representation of a dynamical system. All four presentations maintain the isomorphism of dynamical systems. The advantages of hypergraph representation compared with more simple i-pair representation are the following: • no additional parameters to distinguish among nonisomorphic dynamical systems are

needed; • a standard graph isomorphism algorithm can be used to establish the isomorphism of

multigraphs; • the size of constructed multigraphs is approximately one half of that for Gt .

Mathematica routines for computing all objects constructed in this section are given in the Appendix. Example 18: All representations for one cluster of ocean planetary waves As an example, below all representations are given for one cluster of ocean planetary waves shown in Fig. 3.2, with dynamical systems in real form (the overall results for this case can be found in [124]). Coupling coefficients are denoted as αj for j = 1, 2, 3, . . . ; VH G and EH G are the notations for the set of vertices and the set of edges respectively, of the constructed hypergraph. ,

VH G = {1, 2, 3, 4, 5, 6, 7}, EH G = {2}, {3}, {4}, {7}, {7}, - {1, 2}, {1, 3}, {2, 6}, {4, 5}, {5, 6}, {6, 7}, {1, 3, 4, 5} .

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 1 1 0 0 0 0 1

1 0 0 0 0 1 0 1 0 0 0 0

0 1 0 0 0 0 1 0 0 0 0 1

0 0 1 0 0 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 1 1 0 1

This completes the example.

0 0 0 0 0 0 0 1 0 1 1 0

0 0 0 1 1 0 0 0 0 0 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

A1 = α4 A6 A8 . A2 = α7 A7 A12 . A3 = α10 A9 A12 . A4 = α19 A5 A11 . A5 = α20 A4 A11 . A6 = α5 A1 A8 + α1 A7 A12 . A7 = α8 A2 A12 + α2 A6 A12 . A8 = α6 A1 A6 + α16 A10 A11 . A9 = α11 A3 A12 + α13 A10 A12 . A10 = α17 A8 A11 + α14 A9 A12 . A11 = α21 A4 A5 + α18 A8 A10 . A12 = α9 A2 A7 + α3 A6 A7 + α12 A3 A9 + α15 A9 A10

76

Kinematics: Resonance clusters 6 7 5

1 3 4

T3

1 6

T1

1

1

1

T4

1 4

T5

2

7

T2

5 3

T6

2

T7

Fig. 3.8 Topological structure and multigraph presentation

The form of hypergraph underlies the construction of NR-diagrams given below in Section 3.5. 3.4 Four-wave resonances In this section, we study the geometrical structure of the resonance solution set, taking four-wave resonances of gravity water waves, ω ∼ |k|1/2 , as our main example. Resonance conditions then have the form (2.54) ! 1/2 1/2 1/2 1/2 |k|1 + |k|2 = |k|3 + |k|4 , k1 + k2 = k3 + k4 .

(3.13)

The system (3.13) has been extensively studied (e.g. [111, 117, 118, 125, 154], and others); the overall results are presented below. As in the case of three-wave resonances, each node (m, n) of the integer lattice denotes a wavevector k = (m, n). Four nodes corresponding to a solution of (3.13) can generically be thought of as a quadrangle. Notice that there exist transposed solutions in a four-wave resonance system, e.g. (2.80), which differ only by transposition of some wavevectors. Indeed, each permutation of indexes 1 ↔ 2, 3 ↔ 4 or simultaneous 1 ↔ 3 and 2 ↔ 3 in a solution of (3.13) generates a new solution. But, of course, from a physical point of view all transposed solutions correspond to the same resonance cluster – a quartet – so below they are considered to be one solution. Another specific of a four-wave system is the existence of two different types of quartet, called angle- and scale-resonances [113]. Definition 17. A resonance is called an angle-resonance if it consists of wavevectors with pairwise equal lengths, i.e. |k1 | = |k3 | and |k2 | = |k4 | or |k1 | = |k4 | and |k2 | = |k3 |. Otherwise, a resonance is called a scale-resonance. A resonance cluster consisting of both angle- and scale-resonances is called a mixed cascade.

3.4 Four-wave resonances k-space

50

50

0

0

–50

–50

–100 –100

k-space

100

n

n

100

77

-50

0 m

50

100

–100 –100

–50

0 m

50

100

Fig. 3.9 Geometrical structure of collinear (on the√left) and noncollinear quartets (on the right) in the spectral domain m, n: |k| = | m2 + n2 | ≤ 100. Wavevectors belonging to different resonance quartets are shown in different shades

The importance of this classification is due to the fact that angle-resonances, contrary to scale-resonances, do not generate new wavelengths outside of the initial range of |k| and therefore have different dynamics [113]. Obviously, in a threewave system, all resonances do generate new wavelengths and can be regarded as scale-resonances, which is the main difference between three- and four-wave resonance systems. Scale-resonances Notice that this partitioning of resonances into scale- and angle-types does not coincide with the q-class decomposition case. Indeed, all resonances formed by wavectors from two q-classes are angle-resonances. However, resonances belonging to one q-class can be of both types. Clustering of scale-resonances in the model spectral domain |m|, |n| ≤ 1000 is as follows. The domain contains 230 464 scale-resonances – among them, 213 760 collinear (i.e. all four wave vectors are collinear) and only 16 704 (7.25%) noncollinear resonances [125]. Examples of the geometrical structure of collinear and noncollinear quartets in a smaller domain are shown in Fig. 3.9. However, the coupling coefficient in collinear quartets of gravity water waves is equal to zero and, therefore, they have no dynamical significance [62]. For that reason, below we regard only noncollinear quartets. Noncollinear quartets Example 19: Tridents Tridents, first introduced in [154], are noncollinear scale-resonances of a special form.

78

Kinematics: Resonance clusters

Definition 18. A quartet is called a trident if: (1) there exist two vectors out of four in a quartet, say k1 and k2 , such that k1 ↑↓ k2 , i.e. satisfying (k1 · k2 ) = −k1 k2 ; (2) two other vectors in the quartet, k3 and k4 , have the same length: k3 = k4 ; (3) k3 and k4 are equally inclined to k1 , thus (k1 · k3 ) = (k1 · k4 ) . A parametric series for a trident reads k1 = (a, 0),

k2 = (−b, 0),

k3 = (c, d),

k4 = (c, −d),

(3.14)

with a, b, c, d being known functions of two parameters (see Section 2.2). Parametrization (3.14) corresponds to the tridents with its vectors k1 and k2 oriented along the X-axis and, therefore, we call them axial tridents. There exist also nonaxial tridents, for instance the quartet (49, 49), (−9, −9), (5, 35), (35, 5).

(3.15)

All nonaxial tridents can be obtained from the axial ones (3.14) via rotation by angles, with the rational values of the cosine combined with respective re-scaling in order to obtain integer-valued solutions from rational-valued ones. Example 20: Common quartets Our study of the resonance solution set in the spectral domain |mi |, |ni | ≤ 1000 shows that not all noncollinear scale quartets are tridents. They are further called common quartets, for instance the quartet (990, 180), (128, 256), (718, 236), (400, 200)

(3.16)

is a nontrident quartet. The smallest quartet of this type is shown in Fig. 3.10. The overall number of tridents is 14 848, while the number of nontridents is 1856. First the common quartet (180, 135), (0, 64), (120, 119), (60, 80)

(3.17)

lies in the spectral domain |k| ≤ 225. This means that if we are interested only in large-scale quartets, say, quartets with |k| ≤ 100, the complete set of scaleresonances consists of 1728 quartets, among them 1632 collinear quartets and 96 tridents, but no common quartets [125].

3.4 Four-wave resonances

79

k-space

50

k-space 120

40

100 80

20

n

n

30

60

10

40

0

20

–10 –10

0 0

10

20

m

30

40

50

0

50

100

m

150

Fig. 3.10 Left panel: Nonaxial trident with the smallest possible wavenumbers. Right panel: Nontrident with the smallest possible wave numbers

Fig. 3.11 Clustering of noncollinear quartets

Clusters The structure of clusters consisting of noncollinear quartets is given in Fig. 3.11. Cluster size is the number of quartets belonging to one cluster. An example of a cluster formed by tridents and common quartets is shown in Fig. 3.12. One of the most important characteristics of the resonance set is the wavevector multiplicity introduced in Chapter 2, which describes how many solutions a given wavevector takes part. In Fig. 3.13, in the wavevectors multiplicities are given for noncollinear quartets. It turns-out that 91% of these wavevectors (6720 from overall amount 7384) have multiplicity 1. A cluster of quartets can have quite a complicated structure, for instance in Fig. 3.14 the cluster of four connected quartets is shown, with altogether only seven different wave frequencies [125].

80

Kinematics: Resonance clusters k-space 500

n

0

–500

–1000

–500

0

m

500

1000

Fig. 3.12 A shortest cluster formed by both tridents and common quartets; cluster length is eight, among them six tridents and two nontridents

Fig. 3.13 Wavevectors multiplicity computed for the noncollinear quartets

Fig. 3.14 Example of a cluster consisting of four common quartets. Wavenumbers m, n are shown in the circles as upper and lower numbers. Solid, dashed, dotted– dashed and dotted lines connect the wavevectors forming first, second, third, and fourth quartets of the cluster respectively

3.4 Four-wave resonances

81

Angle-resonances As was mentioned above, angle-resonances do not generate new scales, i.e. new values of |k|, they can only redistribute energy among the modes which were excited initially. However, by observing the one-parametric series of angleresonances given in Section 2.2, (m, n)(t, −n) → (m, −n)(t, n),

t ∈ Z,

(3.18)

we see that the arbitrary wavevector (m, n) takes part in an infinite number of angle-resonances. Notice that this series gives solutions for the arbitrary four-wave system with a dispersion function of the form (m2 +n2 )α , ∀α ∈ Q (though perhaps not all of them). In the computational domain m, n ≤ 103 , there are altogether about 6 · 107 angle-resonances, while the number of noncollinear scale-resonances in the same domain is less than 2 · 104 . Analysis of wavevector multiplicity for angleresonances shows that their multiplicities go very high, indeed the wavevector (1000, 1000) takes part in 11 075 solutions [125]. This indicates that angleresonances should also be taken into account when studying the overall dynamics of the wave field. Mixed cascades An important fact is that the scale- and the angle-resonances are not independent and can form mixed clusters, called mixed cascades, containing both types of resonances. The energy cascade mechanism in such a mixed cluster is presented schematically in Fig. 3.15, upper panel, where the quadrangles S1 and S2 denote scale-resonances so that quartets (V1,1 , ..., V1,4 ) and (V2,1 , ..., V2,4 ) represent scale-resonances, while squares A1 , ..., Ai , .., An represent angle-resonances. If a wave takes part simultaneously in angle- and scale-resonances, then the corresponding nodes of S1 , S2 , and Aj are connected by dashed arrows so that V1,1 = Vn,2 and Vn,1 = V2,3 . Example 21: An example of a mixed cascade, gravity water waves There are many examples of mixed cascades in our resonance set. One is given below, with each quartet being represented as a quadruple (m1 , n1 ), (m2 , n2 ), (m3 , n3 ), (m4 , n4 ). Scale quartet S1 = (−64, −16), (784, 196), (144, 36), (576, 144)

(3.19)

is connected with an angle-quartet An = (−64, −16), (4, 16), (−64, 16), (4, −16),

(3.20)

82

Kinematics: Resonance clusters

Fig. 3.15 (a): Schematic presentation of mixed resonance cascade. (b): Example of mixed cascade. Wavenumbers m, n are shown in the circles as upper and lower numbers. Wave (64,0) takes part in two scale-resonances, both nontridents. The upper number in the circle is m, the lower is n, and thick bold lines drawn between vectors on the same side of equations (3.13). Wave (119,120) takes part in one scale-resonance and in 12 angle-resonances

via modes with wavenumbers V1,1 = Vn,2 = (−64, −16). Angle-quartet An is connected with scale-quartet S2 S2 = (−49, −196), (4, 16), (−36, −144), (−9, −36),

(3.21)

which is further on connected with an angle-resonance An˜ = (−49, −196), (784, 196), (−49, 196), (784, −196),

(3.22)

via the node-vector (−49, −196), [113]. 3.5 NR-diagrams In systems with three-wave resonances, the topological structure of the resonance set is shown as a set of triangles whose vertexes are resonant modes. The mode with highest frequency ω3 is always special in a triad for it has minus in front

3.5 NR-diagrams

83 P

A A P

P

P

P

P

A

P A

A

A

A

A

P

A P

P P

P

P

P

P

P

P

P

P

Fig. 3.16 Examples of clusters in a three-wave system. Upper and middle panels: topological structures of each cluster, lower panel: NR-diagrams of the same clusters

of the interaction coefficient, −Z, in its evolution equation, in contrast to the two other modes. One way to keep this information in the topological representation of a cluster is just to mark the high-frequency mode, ω3 -mode, by the letter A and two other modes by the letter P (Fig. 3.16, upper panel). (Here A and P come from the active and passive mode respectively; the origin of this notation will be explained in Section 4.2). Sometimes the vertex corresponding to the A-mode is shown as a vertex with two out-coming arrows (Fig. 3.16, middle panel). A more elegant and transparent way to represent a cluster graphically is given by the NR-diagram first introduced in [115]. An NR-diagram is a hypergraph with marked arcs, where the (hyper-)vertexes are resonant triads, not modes. Connection types are introduced in the form of two types of half-edges: bold for the A-mode and dotted for the P-mode. Examples of simple clusters met in many wave systems are shown in Fig. 3.16; a more complicated cluster consisting of 11 triads and appearing in the resonance solution set for atmospheric planetary waves [208] is shown in the Fig. 3.17, with its topological structure and corresponding NR-diagram. Given an NR-diagram with N hypervertexes, the corresponding dynamical system is constructed by the coupling of N systems of the form (4.17) and equaling appropriate Bi and Bj . For instance, the dynamical system of the two-triad cluster shown in Fig. 3.16 reads

84

Kinematics: Resonance clusters

Fig. 3.17 11-triads cluster: topological structure (upper panel) and NR-diagram (lower panel)

Fig. 3.18 Topological representation of a butterfly (on the left) and all possible NR-diagrams (on the right)

⎧ ⎪ B˙ = ZB2∗ B3 , B˙ 2 = ZB1∗ B3 , B˙ 3 = −ZB1 B2 , ⎪ ⎨ 1 0 ∗ B6 , B˙ 5 = ZB 0 ∗ B6 , B˙ 6 = −ZB 0 4 B5 , ⇒ B˙ 4 = ZB 5 4 ⎪ ⎪ ⎩ B1 = B4 (PP-connection), B3 = B5 (AP-connection), ! 0 ∗ B6 , B˙ 2 = ZB ∗ B3 , B˙ 1 = ZB2∗ B3 + ZB 5 1 0 ∗ B6 , B˙ 6 = −ZB 0 4 B5 . B˙ 3 = −ZB1 B2 + ZB

(3.23)

(3.24)

4

The system (3.24) is unique, up to the change of indices 1 ↔ 2 and 4 ↔ 5. Notice that the topological structure of a cluster without marked vertexes does not represent a dynamical system uniquely, only when taken together with a complete solution set. We illustrate this with an example of two-triad clusters.

3.5 NR-diagrams

85

Fig. 3.19 Topological representation of a kite (on the left) and all possible NR-diagrams (on the right)

Fig. 3.20 Topological representation of all three-triad clusters (the sign ∅ in the inner triangle shows that this triangle is fictional and does not correspond to a solution)

Fig. 3.21 NR-diagrams for some three-triad clusters shown in Fig. 3.20

It can easily be seen from Figs. 3.18 and 3.19 where for each topological element (called further butterfly and kite respectively) all possible NR-diagrams are shown. According to the connection types within a cluster, we can distinguish three types of butterflies: PP-, PA-, and AA-butterflies and three types of kites: AP-PP-, PP-AA-, and PA-AP-kites, shown in Fig. 3.19 We can also use diagrams to check that a resonance solution set was constructed correctly. Indeed, the cluster shown in Fig. 3.22, on the left, is forbidden due to simple kinematic considerations: if two couples of P -modes coincide, this means that A-modes also coincide, i.e. the cluster is just a triad. The clusters in the middle and on the right are forbidden due to dynamical reasoning: no triad can have two

86

Kinematics: Resonance clusters

Fig. 3.22 Examples of “wrong” NR-diagrams

A-modes. The structure of three-triad clusters is substantially richer as can be seen from the Fig. 3.20. Examples of possible NR-diagrams appearing in threetriad clusters are shown in Fig. 3.21 – one possible diagram for each topological element. We should understand clearly that not all these clusters do appear in every resonance set. For instance, kites appear in resonance sets for oceanic planetary waves and do not appear among resonances of atmospheric planetary waves; butterflies and three-stars appear in both sets, etc. In Table 3.1 some examples are given of clusters appearing in real physical systems, each cluster is represented by an NR-diagram, resonant modes, and dynamical system. In systems with four-wave resonances, we first introduce the notion of one- and two-pairs (their dynamical significance will be explained in Section 4.2). Modes belonging to the one-pair have the same sign of interaction coefficient, while modes belonging to the two-pair have different signs. Correspondingly, connection types within a cluster of quartets are introduced as follows: • E-connection, E for “edge” (via one-pair); • D-connection, D for “diagonal” (via two-pair); • V-connection, V for “vertex” (via one mode).

Later, to construct an NR-diagram we will need two types of vertices: squares for scale-resonances and circles for angle-resonances. Now the V-connection is shown as a single bold line, the E-connection as a double bold line, and the D-connection as a dashed line. Again, the dynamical system for each cluster can be written out explicitly, for instance, for an E-scale-scale cluster consisting of the two-quartet cluster shown in Fig. 3.23 it reads ⎧ ⎪ B˙ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ ⎨B3 B˙ 5 ⎪ ⎪ ⎪ ⎪ B˙ 7 ⎪ ⎪ ⎪ ⎪ ⎩B 1

= ZB2∗ B3 B4 ,

B˙ 2 = ZB1∗ B3 B4 ,

= Z ∗ B4∗ B1 B2 , B˙ 4 = Z ∗ B3∗ B1 B2 , 0 ∗ B7 B8 , B˙ 6 = ZB 0 ∗ B7 B8 , ⇒ = ZB 6 5 ∗ ∗ ∗ 0 0 ˙ = Z B B5 B6 , B8 = Z B ∗ B5 B6 , 8

= B5 , B2 = B6 ,

7

(3.25)

3.5 NR-diagrams

87

Table 3.1 Examples of resonance clusters found among atmospheric planetary waves, resonance conditions are given by (2.94). For each cluster its NR-diagram and resonant modes are shown in the 1st column, while the explicit form of corresponding dynamical system is given in the 2nd column

a: (9,116)(105,231)(114,208) b: (85,187)(29,376)(114,208) B3|a = B3|b

∗ B , B ∗ B˙ 1|a = Za B2|a 3|a ˙ 1|b = Zb B2|b B3|a , ∗ ∗ B , B˙ 2|a = Za B1|a B3|a , B˙ 2|b = Zb B1|b 3|a B˙ 3|a = −Za B1|a B2|a − Zb B1|b B2|b

a: (5,24)(2,15)(7,20) b: (6,18)(7,20)(13,19) B3|a = B2|b

∗ B , B B˙ 1|b = Zb B3|a 3|b ˙ 3|b = −Zb B1|b B3|a , ∗ ∗ B , B˙ 3|a = Zb B1|b B3|b , B˙ 2|a = Za B1|a 3|a ∗ B B˙ 3|a = −Za B1|a B2|a + Zb B1|b 3|b

a: (2,6)(3,8)(5,7) b: (2,6)(4,14)(6,9) B1|a = B1|b

∗ B ∗ B˙ 1|a = Za B2|a 3|a + Zb B2|b B3|b , ∗ ∗ B , B˙ 2|a = Za B1|a B3|a , B˙ 2|b = Zb B1|a 3|b B˙ 3|a = −Za B1|a B2|a , B˙ 3|b = −Zb B1|a B2|b

a: (18,186)(417,527)(435,464) b: (210,598)(225,399)(435,464) c: (90,340)(345,527)(435,464) B3|a = B3|b = B3|c

B˙ 1|a B˙ 1|b B˙ 1|c B˙ 3|a

∗ B , B ∗ = Za B2|a 3|a ˙ 2|a = Za B1|a B3|a , ∗ B , B ˙ 2|b = Zb B ∗ B3|a , = Zb B2|b 3|a 1|b ∗ ∗ B , = Zc B2|c B3|a , B˙ 2|c = Zc B1|c 3|a = −Za B1|a B2|a − Zb B1|b B2|b − Zc B1|c B2|c

a: (125,720)(146,927)(271,812) b: (145,840)(126,783)(271,812) c: (271,812)(270,810)(541,811) B3|a = B3|b = B1|c

B˙ 1|a B˙ 1|b B˙ 3|c B˙ 3|a

∗ B , B ∗ = Za B2|a 3|a ˙ 2|a = Za B1|a B3|a , ∗ B , B ∗ = Zb B2|b 3|a ˙ 2|b = Zb B1|b B3|a , ∗ B , = −Zc B3|a B2|c , B˙ 2|c = Zc B3|a 3|c ∗ B = −Za B1|a B2|a − Zb B1|b B2|b + Zc B2|c 3|c

a: (154,602)(231,902)(385,737) b: (226,903)(5,806)(231,902) c: (231,902)(22,615)(253,860) B2|a = B3|b = B1|c

B˙ 3|a B˙ 1|b B˙ 2|c B˙ 2|a

∗ B , = −Za B1|a B2|a , B˙ 1|a = Za B2|a 3|a ∗ B , B ∗ = Zb B2|b 2|a ˙ 2|b = Zb B1|b B2|a , ∗ B , = Zc B2|a B˙ 3|c = −Zc B2|a B2|c , 3|c ∗ ∗ B = Za B1|a B3|a + Zc B2|c 3|c − Zb B1|b B2|b .

a: (238,713)(303,909)(541,805) b: (303,909)(261,594)(546,714) c: (8,287)(303,909)(311,819) B2|a = B1|b = B2|c

B˙ 3|a B˙ 2|b B˙ 1|c B˙ 2|a

∗ B , = −Za B1|a B2|a , B˙ 1|a = Za B2|a 3|a ∗ B , B = Zb B2|a 3|b ˙ 3|b = −Zb B2|a B2|b , ∗ B , B = Zc B2|a 3|c ˙ 3|c = −Zc B1|c B2|a , ∗ ∗ B ∗ = Za B1|a B3|a + Zb B2|b 3|b + Zc B1|c B3|c .

88

Kinematics: Resonance clusters

Fig. 3.23 NR-diagrams of clusters in appearing in four-wave systems (left to right and up to down): primary angle-resonance, primary scale-resonance, mixed cascade consisting of five quartets with one E-angle-scale, one ED-scale-scale and two V-scale-angle connections; pure scale-resonance cluster consisting of four quartets shown in Fig. 3.14; E-scale-scale cluster

⎧ 0 ∗ B7 B8 , B˙ 2 = ZB ∗ B3 B4 + ZB 0 ∗ B7 B8 , ⎪ B˙ 1 = ZB2∗ B3 B4 + ZB ⎪ 2 1 1 ⎪ ⎨ ∗ ∗ ∗ ∗ ˙ ˙ B3 = Z B4 B1 B2 , B4 = Z B3 B1 B2 , ⎪ ⎪ ⎪ ⎩B˙ = Z 0∗ B ∗ B1 B2 , B˙ 8 = Z 0∗ B ∗ B1 B2 . 7 7 8

(3.26)

Here the E-connection has been realized via edges (B1 , B2 ) (first quartet) and (B5 , B6 ) (second quartet). The resulting system (3.26) is unique up to permutation of indexes 1 ↔ 2 and 5 ↔ 6. Superficially, NR-diagrams for resonance clusters look somewhat similar to the Feynman diagrams known in quantum mechanics and were first introduced by Wyld [254] for the study of turbulence. The differences are as follows: • In a Feynman diagram each vertex represent a particle, which corresponds to a single

wave in the topological representation of a cluster, while in the NR-diagram vertexes are primary clusters – resonant triads or quartets. • A Feynman diagram represents a term in the Wick’s expansion of the perturbative matrix, i.e. the sum of all Feynman diagrams represents the possible interactions of a given particle with other particles. In contrast, one NR-diagram represents completely a resonance cluster. • A Feynman diagram does not allow us to compute the amplitudes of the scattering process; it only gives a contribution corresponding to one term in the perturbation expansion. However, the NR-diagram allows us to reconstruct a uniquely dynamical system, where the solutions are the amplitudes of resonantly interacting modes.

3.6 What is beyond kinematics? Let us summarize what we have learned so far. In Chapter 1, we described the class of nonlinear evolutionary dispersive PDEs, possessing resonance solutions. We also introduced the notion of nonlinear resonance and have shown that conditions

3.6 What is beyond kinematics?

89

of resonance can be written out explicitly as an algebraic system of equations on the indexes (m, n) of Fourier harmonics or normal modes corresponding to given boundary conditions. In Chapter 2, we have learned how to compute these indexes (m, n) and in Chapter 3 how to construct resonance clusters, represent them in the form of NR-diagrams, and write out the explicit form of the dynamical system corresponding to a cluster. Our next step is to study the solutions of these systems, i.e. the dynamics of resonance clusters.

4 Dynamics

But the drawings did not help, and the further it went, the more confusing and incomprehensible the poet’s statement became. M. Bulgakov Master and Margarita

4.1 An easy start Let us regard the primary dynamical system for a triad in canonical variables B˙ 1 = ZB2∗ B3 ,

B˙ 2 = ZB1∗ B3 ,

B˙ 3 = −ZB1 B2 ,

(4.1)

with two constants of motion in the Manley–Rowe form [164]: I13 = |B1 |2 + |B3 |2 ,

I23 = |B2 |2 + |B3 |2 .

(4.2)

Obviously, the linear combination of I13 and I23 I12 = I13 − I23 = |B1 |2 − |B2 |2

(4.3)

also is a constant of motion. However, any two of I13 , I23 , I12 are functionally independent and can be taken as conservation laws of (4.1). If the amplitudes Bj are real-valued functions, we have a system of three variables, and two conservation laws are enough to find a solution to (4.1). Indeed, direct substitution of (4.2) into (4.1) allows us to obtain three independent ODEs, one for each amplitude, and solve them separately in Jacobian elliptic functions. The situation becomes more complicated if the amplitudes Bj are complexvalued functions. Now we have six variables: the real and imaginary parts of each amplitude or, equivalently, the modulus and phase of each amplitude. It looks as though we need three more conservation laws for (4.1) to be integrable. In fact, it turns out that one more additional conservation law – namely, the Hamiltonian HT – is enough to provide integrability of (4.1). 90

4.1 An easy start

91

Indeed, let us rewrite (4.1) in the standard amplitude–phase presentation Bj = Cj exp(iθj ): ⎧ ⎪ C˙ 1 = ZC2 C3 cos ϕ12|3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨C˙ 2 = ZC1 C3 cos ϕ12|3 , ⎪ C˙ 3 = −ZC1 C2 cos ϕ12|3 , ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩ϕ˙12|3 = −Z HT C −2 + C −2 − C −2 , 1 2 3

(4.4)

where ϕ12|3 = θ1 + θ2 − θ3 is called the dynamical phase of a triad [31]. Notice that conservation laws (4.2) do not change their form in the new variables: I23 = C22 + C32 ,

I13 = C12 + C32 .

(4.5)

The system can now be easily rewritten as d C32 d C22 d C12 = =− = 2ZC1 C2 C3 cos ϕ12|3 , dt dt dt

(4.6)

with Hamiltonian HT HT = C1 C2 C3 sin ϕ12|3 .

(4.7)

Thus, (4.1) possesses one more conservation law and it can be reduced to the form (4.4), which does not depend on the individual phases θ1 , θ2 , θ3 , but only on their combination (4.8) ϕ12|3 = θ1 + θ2 − θ3 corresponding to the resonance conditions chosen. This way (4.1) is rewritten as (4.4), with four variables C1 , C2 , C3 , ϕ12|3 and three conservation laws (4.5)– (4.7), and again can be solved in Jacobian elliptic functions. To cut out the tedious formulas, we give here as an example the general expressions for C12 , not for C1 , in the case of real amplitudes:  C12 (t)

= dn

2

I13 t · Z I13 , I13 I23 

(4.9)

and in the case of complex amplitudes: 

   C12 (t) = α1 + α2 α3 · dn2 α4 · t · Z I13 , α5 − α6 I13 with coefficients α1 , ..., α6 being functions of I13 , I23 , and HT .

(4.10)

92

Dynamics

In order to get the complete solution to (4.4), the solutions for amplitudes (4.10) should be accompanied by the solution to the equation for corresponding dynamical phase # $ 1 1 1 (4.11) + − 2 . ϕ˙12|3 = −HT C12 C22 C3 The form of the Hamiltonian (4.7) has been known in plasma physics for at least 40 years [232] and most probably had been known even before that in nonlinear optics. The importance of the dynamical phase was also recognized in those times (see, for instance [251, 252] and others), but the problem was not solved analytically till recently. The profound analytical and numerical study of the dynamical phase behavior is given in papers [156, 157, 158], where pulsation and precession of the elastic pendulum have been studied. The classical form of the solution is given in Jacobian elliptic functions sn, dn, and cn (see e.g. [249, 152, 122], etc.), and a novel and beautiful representation of the solution in terms of Weierstrass elliptic functions has been recently obtained in [158]. Similarly, consider the primary dynamical system in a four-wave resonance system ⎧ ⎪ B˙ 1 = ZB2∗ B3 B4 , ⎪ ⎪ ⎪ ⎪ ⎨B˙ 2 = ZB ∗ B3 B4 , 1 (4.12) ∗B B , ⎪ ˙ ⎪ = −ZB B 3 1 2 ⎪ 4 ⎪ ⎪ ⎩˙ B4 = −ZB ∗ B1 B2 . 3

Again, it can be rewritten in amplitude–phase variables d C32 d C22 d C42 d C12 = =− =− = 4C1 C2 C3 C4 sin ϕ12|34 , dt dt dt dt

(4.13)

where the dynamical phase of a quartet ϕ12|34 = ϕ1 + ϕ2 − ϕ3 − ϕ4 ,

(4.14)

is the only combination of phases that affects the quartet dynamics. Its Hamiltonian, together with three independent Manley–Rowe integrals, say I13 = C12 + C32 ,

I14 = C12 + C42 ,

I23 = C22 + C32 ,

(4.15)

allow us to express four amplitudes Cj in terms of just one of them. Therefore, the phase space of the quartet becomes two dimensional as is the case for a triad, and can be described in terms of two variables, say C1 and ϕ12|34 . The general answer

4.2 Decay instability

93

can be given again in terms of Jacobian elliptic functions, though the analytical expressions are more involved (see [223] for details; the corresponding dynamical system is regarded in a general form (1.127)). Below we regard mostly the clusters formed by triads and use notation ϕ instead of ϕ12|3 , if appropriate, to simplify the resulting formulas. In Chapter 4 we study dynamical systems appearing due to resonance clustering. 4.2 Decay instability Decay instability of a triad Before proceeding with the study of time evolution of resonance clusters, we would like to show what dynamical characteristics are hidden in the notion of A- and P-modes (three-wave system) and one- and two-pairs (four-wave system) introduced at the end of the previous chapter. Indeed, simple considerations based on the form of conservation laws of primary dynamical systems (1.125) and (1.126) allow us to make some conclusions on the decay instability of resonance clusters. Decay instability is an important physical phenomenon known by different names in various areas of physics: parametric instability of the Kelvin–Helmholz flow [130], the Anderson–Suhl instability of spin waves [6], the Oraevsky–Sagdeev instability of plasma waves [189], etc. In [88], Hasselmann first formulated the criterion for nonlinear wave stability for the general form of dynamical systems (1.125) and (1.126), deducing it directly from the equations of motion. His formulation (in our notation) is as follows: The nonlinear coupling between two infinitesimal components 1 and 2 and a finite component 3 whose wavenumbers and frequencies satisfy resonance conditions ω1 ± ω 2 = ω 3

(4.16)

is unstable for sum interaction and neutrally stable for difference interaction. An elementary way to demonstrate this is to transform resonant dynamical systems into a form that yields constants of motion in the Manley–Rowe presentation [163] and rewrite them in the canonical form B˙ 1 = ZB2∗ B3 ,

B˙ 2 = ZB1∗ B3 ,

B˙ 3 = −ZB1 B2 .

(4.17)

Looking at the form of the Manley–Rowe constants (4.2), we conclude immediately that if B1 (t = 0) # B2 (t = 0) and B1 (t = 0) # B3 (t = 0), then I23 (t = 0) # I13 (t = 0). Integrals of motion are independent of time, therefore I13 # I23 at each moment of time, and hence |B1 (t)|2 # |B2 (t)|2 .

94

Dynamics

Moreover, |B1 (t)|2 # |B3 (t)|2 at every moment. Indeed, the assumption |B1 (t)|2  |B3 (t)|2 yields I13 $ I23 and this is not always the case. This means that the ω1 -mode, being the only one substantially excited at t = 0, cannot share its energy with two other modes in a triad, and similar considerations remain true for the ω1 -mode. This is the reason why they are referred to as P-modes – here P comes from passive. The situation changes if the ω3 -mode is initially excited. In this case, the P-mode amplitudes will grow exponentially, |B1 (t)|, |B2 (t)| ∼ exp[ |ZB3 (t = 0)|t], until all the modes have comparable magnitudes of amplitude. Being initially excited, the ω3 -mode is capable of sharing its energy with two P-modes within a triad and is called the active mode, or A-mode. All connections within an arbitrary resonance cluster of triads can be then described in terms of the AA-, AP-, and PP-connection types first introduced in [123]. Decay instability of a quartet A similar approach for a four-wave system (1.126), whether based on Hasselman’s deduction or on the Manley–Rowe form of constants of motion I13 = B12 + B32 ,

I14 = B12 + B42 ,

I23 = B22 + B32 ,

(4.18)

shows that for this case no general criterion of stability can be formulated. However, some particular cases can be treated. Indeed, if any one mode is initially excited, say the ω1 -mode, then I13 ≈ B12 ,

I14 ≈ B12 ,

I23 ≈ 0

(4.19)

and amplitudes B2 , B3 , B4 cannot grow, i.e. energy transfer is essentially suppressed. If two modes are initially excited, stability depends on the choice of the couple. Indeed, for an arbitrary two-pair, say the ω1 -mode and ω3 -mode, energy transport is still suppressed while I13 ≈ B12 ,

I14 ≈ 0,

I23 ≈ B32

(4.20)

and the ω2 -mode and ω4 -mode do not grow. On the other hand, if a one-pair couple is excited initially, equations (4.18) do not globally suppress the interactions, and the excitation of one of these couples yields effective or suppressed energy transfer, depending on the details of the disturbances. This can easily be seen after rewriting (4.18) as I13 − I14 = B32 − B42 , I14 + I23 =

B12

I13 − I23 = B12 − B22 , + B42

+ B22

+ B32 .

(4.21) (4.22)

4.2 Decay instability

95

In a particular case, when some one-pair is initially infinitesimal and enters twice in the interaction, resonance conditions (1.124) turn into k1 + k2 = 2k3 ,

ω1 + ω2 = 2ω3 ,

(4.23)

and the three-wave Hasselmann’s criterion of instability can be used, of course. Reduction types It is clear from the dynamical considerations in the previous section that any triad, connected to the rest of a generic cluster via a P P -connection cannot accept energy from the cluster if initially its A-mode is not excited. Thus connections within a cluster involving only P -modes are not penetrative for energy in any direction. Therefore, from the dynamical viewpoint it might be useful to divide large clusters into “almost separated” subclusters cutting off all PP-connections. This procedure of cluster reduction is called PP-reduction. Generic clusters with and without PP-connections are called P P -reducible or PP-irreducible accordingly. Whether or not a cluster is PP-reducible can be seen immediately from the form of its NR-diagram: each time a connection formed by two dashed half-arcs occurs, we can “cut” it off in the middle (shown as a small circle). For instance, the four-triad chain shown in Fig. 3.15 is PP-reducible into two isolated triads and one AP -butterfly; PP-reduction of the 11-triad cluster shown in Fig. 3.16 yields three isolated triads and one P P -irreducible eight-triad cluster, and so on. Notice that PP-reduction does not necessarily diminish the size of a cluster: the three-triad cluster shown in Fig. 3.20 (bottom on the left) is P P -reducible, but the number of triads stays the same. However, the number of connections is reduced and therefore the resulting dynamical system has a more simple form. In a cluster of quartets, V-reduction is possible. This is an important fact because it allows in particular to cut off angle-resonances that are numerous. For instance, as we can see from (3.18), if the dispersion function has the form (m2 + n2 )α , ∀α ∈ Q, an arbitrary wavevector (m, n) participates in an infinite number of resonances described by a one-parametric series. This means that, in an arbitrary scale-quartet, each mode is connected with an infinite number of angleresonances of the form (3.18), while in a finite domain its size defines the number of possible connections. However, V-reduction allows us to regard it as an isolated quartet. Information about cluster reducibility can readily be obtained from the form of its NR-diagram. Indeed, in a triad cluster all connections formed by two dashed half-edges can be cut off, while in a quartet cluster connections shown as a single bold line can be omitted.

96

Dynamics

Of course, clusters can be regarded as reducible not for generic initial conditions but only for those discussed above (see example in Section 4.8). 4.3 A triad In this section, the results for a complex triad are briefly presented on the dynamical invariant and solutions for amplitudes [30, 116], dynamical phase [31], and degenerate initial conditions [108]. Time-dependent conservation laws Definition 19. Consider a general N -dimensional system of autonomous evolution equations of the form dx i (t) = i (x j (t)), dt

i = 1, . . . , N.

(4.24)

 ∂ d f (x i (t), t) = f + i f,i = 0 dt ∂t

(4.25)

Any scalar function f (x i , t) that satisfies

is called a conservation law [187]. Here notation f,i ≡ ∂f/∂x i

(4.26)

is used. Obviously, this definition gives us two types of conservation law: time-independent conservation law is of the form f (x i ) and time-dependent law of the form f (x i , t). To keep in mind the difference between these two types of conservation laws, we refer to the first type as just a conservation law, and to the second type as a dynamical invariant [30]. Definition 20. The dynamical system (4.24) is called integrable if it has N functionally independent dynamical invariants. Obviously, if (4.24) possesses (N −1) functionally independent conservation laws, then it is constrained to move along a one-dimensional manifold, and the way it moves is dictated by one dynamical invariant. Accordingly, the dynamical invariant can be computed if (N − 1) conservation laws and the explicit form of (4.24) are known. In this case, (4.24) will be integrable . It follows from the theorem below that in many cases the existence of only (N − 2) conservation laws is enough for integrability of (4.24).

4.3 A triad

97

Theorem 10. Let us assume that the system (4.24) possesses a standard Liouville volume density ρ(x i ) : (ρi ),i = 0, and (N − 2) functionally independent conservation laws, H 1 , . . . , H N−2 . Then a new conservation law in quadratures can be constructed, which is functionally independent of the original ones, and therefore the system is integrable. This theorem was first formulated in [30] and proven there for the case of N = 2; the complete proof follows from the existence of a Poisson bracket for (4.24) under the assumptions of the theorem. The proof is constructive and allows us to find the explicit form of a new conservation law for our dynamical systems. In the examples below, the number N corresponds to the effective number of degrees of freedom, i.e. the dynamical phase ϕ12|3 is counted, not the individual phases θ1 , θ2 , θ3 .

Dynamical invariant for a triad The dynamical system (4.1) corresponds to N = 4 and the direct application of Theorem 10 yields a conservation law

HT = Im B1 B2 B3∗ ,

(4.27)

which is, of course, the Hamiltonian of (4.1) and can be obtained directly. The time-dependent dynamical invariant has the form (first obtained in [30]):

1/2  1/2  −v R3 −R2 F arcsin RR33−R , R3 −R1 2 S =Zt − 2

. 2 1/4 21/2 (R3 − R1 )1/2 I13 − I13 I23 + I23

(4.28)

Here F is the elliptic integral of the first kind, R1 < R2 < R3 are the three real roots of the polynomial

x 3 + x 2 = 2/27 − 27HT2 − (I13 + I23 )(I13 − 2I23 ) 

3/2 2 2 × (I23 − 2I13 ) /27 I13 − I13 I23 + I23 and

 v = |B1 | − 2I13 − I23 + 2

2 I13

2 − I13 I23 + I23

1/2 1 3

(4.29)

(4.30)

98

Dynamics

is always within the interval [R2 , R3 ], which contains zero. As v is time dependent, v(t), the period of motion is defined by the time it takes v(t) to go from R2 to R3 and then back to R2 and can be computed from the above equation. The trigonometric representation (Casus Irreducibilis [239]) of the three real roots R1 < R2 < R3 , ⎧ 1 ⎪ ⎪ R1 = − (1 + 2 cos(α/3 − π/3)), ⎪ ⎪ ⎪ 3 ⎪ ⎨ 1 (4.31) R2 = − (1 + 2 cos(α/3 + π/3)), ⎪ 3 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩R3 = − (1 − 2 cos(α/3)), 3 together with new variables ρ and α ∈ [0, π ] defined as

and

ρ = I23 /I13

(4.32)

−2 + 3 ρ + 3 ρ 2 − 2 ρ 3 I13 3 − 27 HT 2 cos α =

3/2 3 2 1 − ρ + ρ2 I13

(4.33)



yield the following expression for the dynamical invariant S, [116]: S(t, C1 , C2 , C3 , ϕ) = 2 t−

2(t − t0 ) + T 2T

4

3 T +

2(t−t0 )+T T

5

(−1) 2 K(μ)

T

F(μ).

(4.34)

Here · is the floor function, 6 α

2

⎞ 7 2 2 2 2 7 cos 3 Z T C3 − C 2 − C 1 F(μ) = F ⎝arcsin 8 √ , μ⎠ (4.35) α π + 12 μ K(μ)2 3 cos 3 + 6 ⎛

and K(μ) are complete elliptic integrals of the first kind with modulus μ = cos and period

α 3

+

α π  π / cos − , 6 3 6

√ 1 2 3 4 K (μ) T =

1 "

√ , Z 1 − ρ + ρ 2 4 cos α3 − π6 I13

(4.36)

(4.37)

4.3 A triad

99

while the parameter t0 is defined by the initial conditions: t0 = sign(cos ϕ(0))

T F(μ)|t=0 . 2 K(μ)

(4.38)

Solutions for amplitudes To simplify the presentation, the solution of (4.4) is given for the amplitude squares: ⎧   ⎪ 2K(μ) 2 2 (t − t0 ) ⎪ 2 ⎪C1 (t) = −μ ,μ sn 2 K(μ) ⎪ ⎪ ZT T ⎪ ⎪

 ⎪ I13 α  ⎪ ⎪ 2 cos ⎪ + 2 − ρ + 2 , 1 − ρ + ρ ⎪ ⎪ 3 3 ⎪ ⎪  2 ⎪ ⎪ ⎪ ⎨C 2 (t) = −μ 2K(μ) sn2 2 K(μ) (t − t0 ) , μ 2 ZT T (4.39)

α   ⎪ I13 ⎪ 2 ⎪ + 2ρ − 1 + 2 1 − ρ + ρ cos , ⎪ ⎪ 3 3 ⎪ ⎪   ⎪ 2 ⎪ (t − t0 ) ⎪ 2 (t) = μ 2K(μ) ⎪ ⎪ C ,μ sn2 2 K(μ) ⎪ 3 ⎪ ZT T ⎪ ⎪

 ⎪ I13 α  ⎪ ⎩ + ρ + 1 − 2 1 − ρ + ρ 2 cos , 3 3 where sn(·, μ) is a Jacobian elliptic function and t0 is defined by the initial conditions. Using standard identities for Jacobian elliptic functions, sn2 + cn2 = 1 and sn2 + dn2 = 1, we choose for presentation any of the functions sn, cn, dn. An easy check shows that if HT = 0, the period reads 2K(μ) T˜ = √ Z I13

(4.40)

with μ = ρ, while the solution takes a very familiar form, which can be found in many publications (e.g. [152, 249] and others): ⎧

√ ⎪ C˜ 12 (t) = dn2 (t − t0 ) Z I13 , ρ I13 , ⎪ ⎨

√ (4.41) C˜ 22 (t) = cn2 (t − t0 ) Z I13 , ρ I23 , ⎪ ⎪

√ ⎩ ˜2 C3 (t) = sn2 (t − t0 ) Z I13 , ρ I23 . The

triad dynamics is periodic and is shown in Fig. 4.1 in coordinates 2 typical C1 , ϕ, t . However, even in the case of real amplitudes C˜ j , the time evolution of the modes can become nonperiodic for some initial conditions [108, 152], which are further referred to as degenerate initial conditions.

100

Dynamics

p /4

0

j (t) p /2

0

3 p /4

0.5

2 1 C1 (t)

0

1.5

3p /4

j(t) p /2

3p /4

p

0

0.5 1

C12(t) 1.5 2

0.5 1

C12(t) 1.5 2

5

2

2

p /4

j (t) p /2

2.5

t 0

0

t

–2

–2.5 –5

p /4

0

j (t) p /2

p 0

3p /4

0.5 2 1 C1 (t) 1.5 2

4

t

p /4

0

p 0

5 2.5

2 t

0

0 –2.5

–2

–5

Fig. 4.1 Level surface of Hamiltonian HT , level surface of dynamical invariant

S, solution trajectory C12 (t), ϕ(t) and their combined plot

Degenerate initial conditions To understand how the period can become infinite, let us recall the definition of the Jacobian elliptic functions sn, cn, dn, [3]. If 

φ

u= 0

dθ (1 − μ sin2 θ)1/2

(4.42)

,

then sn(u) = sin φ,

cn(u) = cos φ,

dn(u) = (1 − μ sin2 φ)1/2

(4.43)

and they are periodic in u with periods proportional to the complete elliptic integral  K= 0

If μ = 1,

 K= 0

π/2

π/2

dθ (1 − μ sin2 θ)1/2

.

1 1 + sin φ π/2 dθ = ln | =∞ cos θ 2 1 − sin φ 0

(4.44)

(4.45)

4.3 A triad

and the functions



1 1 + sin φ ln cn(u) = dn(u) = sin 2 1 − sin φ

101

(4.46)

decrease from 1 to 0 when u is increasing from 0 to 1. At the same time, the function sn(u) grows from 0 to 1. It follows from (4.36) that μ=1

α=0

if

(4.47)

and together with (4.32) and (4.33) we conclude that initial conditions are degenerate if the following equation holds: 1=

3 + 3I I 2 + 3I I − 2I − 27H 2 −2I13 23 13 23 13 23 T 3 2[1 − I23 /I13 + (I23 /I13 )2 ]3/2 I13

.

(4.48)

In the case of real amplitudes HT = 0, the condition of degeneracy takes a very simple form in canonical variables: 1=μ=ρ=

I23 |B2 |2 + |B3 |2 = . I13 |B1 |2 + |B3 |2

(4.49)

To demonstrate what this means in terms of the physical variables Aj , we regard as an example atmospheric planetary waves. Example 22: Atmospheric planetary waves Dynamical equations in physical variables for this case are given by (1.101) and modulus μ reads μ=

A230 N3 /(N2 − N1 ) + A220 N2 /(N3 − N1 ) A230 N3 /(N2 − N1 ) + A210 N1 /(N3 − N2 )

.

(4.50)

Let us denote as Ej 0 = Nj A2j 0 the initial energies of the modes Aj , j = 1, 2, 3, and introduce new variables x1 , x2 : E10 = x1 E30 ,

E20 = x2 E30 ,

(4.51)

so that x1 and x2 define the fraction of the A-mode’s initial energy E30 that is initially contained in each of the P-modes. In these variables, the modulus μ takes the form μ=

x2 (N2 − N1 ) + (N3 − N1 ) N3 − N2 · x1 (N2 − N1 ) + (N3 − N2 ) N3 − N1

(4.52)

102

Dynamics

and the case μ = 1 implies x2 x1 = . N3 − N2 N3 − N1

(4.53)

The degeneracy condition (4.53) does not depend on the total initial energy of a triad. The crucial parameter is the initial energy distribution among the modes of a triad. For any fixed triad, Nj are constants and (4.53) describes the line in the plane (x1 , x2 ), where the integral K diverges and accordingly the period of energy oscillation becomes infinite. On the other hand, the magnitude of the period depends, of course, on the total initial energy of a triad. This implies that the formula for the period (4.40) is reliable for practical purposes only if small variations in x1 , x2 due to small errors in the initial data lead to small deviations in μ and correspondingly in K and T . If a minor change of the initial conditions yields a change of order of K, the formula (4.40) is useless. This happens only in a narrow domain of the magnitudes of μ, close to 1, because K(μ) changes slowly enough: from π/2 to 3.69, while μ grows from 0 to 0.99 [3]. To estimate the size of the “sensitive” neighborhood  of the degenerate line x1 /x2 , we can consider the power series representation of K 9 :  2  2  2 n=∞ 1 3 5 π  π 13 1 1+ K= pn μn = μ+ μ2 + μ3 + · · · . 2 2 2 24 246 n=0

(4.54) Its coefficients can be expressed in terms of the -function and estimated from above using the Stirling formula: 

(2n + 1) pn = (n + 1)2 22n

2 0 and are therefore of minor importance. We demonstrate now that this is not quite accurate, making use of the Thue–Siegel– Roth theorem [215, 162]: Theorem 11 (Thue–Siegel–Roth theorem). If the algebraic numbers α1 , α2 , . . . , αs are linearly independent with 1 over Q, then for any ε > 0 we have |p1 α1 + p2 α2 + · · · + ps αs − p| > cp −s−ε for all p, p1 , p2 , . . . , ps ∈ Z with p = maxi |pi |. The nonzero constant c has to be constructed for every specific set of algebraic numbers separately. This theorem thus defines the minimal resonance width δω > 0 necessary for approximate interactions to start. A very enlightening introduction to the ideas of Diophantine approximations and the measure of irrationality of a number can be found in [161]. Let us regard a couple of simple examples showing how to apply this theorem for a given dispersion function. Example 23: Global lower boundary for gravity water waves For four-wave resonances of gravity water waves, we have ω ∼ (m2 + n2 )1/4

(6.2)

and Theorem 11 yields in particular that |ω1 ± ω2 ± ω3 ± ω4 | > 1

(6.3)

if at least one of ωi is not a rational number. As has been shown in Section 2.2, all integer solutions in this case have one of two forms: (I) ωi = γi q 1/4 , ∀i = 1, 2, 3, 4 1/4 1/4 or (II) ω1 = ω3 = γ1 q1 , ω2 = ω4 = γ2 q2 , q1 = q2 . This means that Theorem 11 gives a big resonance width δω = 1 for all quasi-resonances except those formed

148

Wave turbulent regimes

by vectors of the class Cl1 . This last case obviously possesses a similar estimate: if wavevectors belong to the class Cl1 , all ωj are integers and their linear combination is also an integer, i.e. δω ≥ 1. Example 24: Global lower boundary for sound waves

√ For three-wave resonances of sound waves, we have ω ∼ m2 + n2 and, similar to the previous example, this yields |ω1 + ω2 − ω3 | > 1 if at least one of ωi is not a rational number. As follows from Section 2.2, ωi is a rational number only if the vector (mi , ni ) belongs to the class Cl1 . As in the previous example, δω ≥ 1 for the class Cl1 . The low boundaries found in Examples 23 and 24 depend only on the form of the dispersion function, not on the wavenumbers’ range. This is the reason why they are called global lower boundaries. It is important to realize that for some dispersion functions – for instance, rational – only a local lower boundary exists, i.e. the corresponding boundary depends on the size of the spectral domain. We illustrate this with the example below. Example 25: Local lower boundary for atmospheric planetary waves Let us regard the dispersion function for atmospheric planetary waves in the following form: ω ∼ m/[n(n + 1)]. Then 1 n1 n2 n3 (n1 + 1)(n2 + 1)(n3 + 1) 1 ≥ [maxj (nj + 1)]6

|ω1 ± ω2 ± ω3 | ≥

(6.4)

and 1 →0 [maxj (nj + 1)]6

if nj → ∞.

(6.5)

On the other hand, a local low boundary always exists, which is defined by the spectral domain D under consideration, D = {(m, n) : 0 < |m|, |n| ≤ T < ∞}. Indeed, let us define T = minp p , where < p

p

p < p = < ω k 1 ± ω k 2 ± · · · ± ω k 4 < ,

p p

p kj = mj , nj ∈ D,

for ∀j = 1, 2, 3, 4, and p

p

p

ω k1 ± ω k2 ± · · · ± ω k4 = 0 ∀p,

(6.6)

(6.7)

(6.8)

6.2 Quasi-resonances vs approximate interactions

149

where the index p runs over all wavevectors in D, i.e. p ≤ 4T 2 . So defined, p obviously is a nonzero number as the minimum of the finite number of nonzero numbers and T is the minimal resonance width. Usually, the step of numerical schema 0 < δ  1 is taken as a parameter characterizing quasi-resonant regimes, and relations between δ, D , and δω describe all these regimes. For instance, if D > , any chosen δ0 > D will allow some number of quasi-resonances, say N(δo > D ) , and for any δ > δ0 , N(δ > D ) ≥ N(δo > D ) . On the other hand, if δ is decreasing to D , δ → D + 0, the number of quasi-resonances reaches some constant level Nmin , Nmin = N(δ = D ) . If δ is increasing to D , δ → D − 0, the number of quasiresonances is Nmin = const. We can see that the standard wording “wave interactions” is misleading in this context, and also the notion of quasi-resonance has to be qualified. The notions of (exact) resonance, quasi-resonance, and approximate interaction can be defined by the radius R = cp −s−ε (6.9) from Theorem 11 and resonance width δω , as follows: exact resonances are solutions of (6.1) with δω = 0,

(6.10)

quasi-resonances are solutions of (6.1) with 0 < δω < R,

(6.11)

approximate interactions are solutions of (6.1) with R ≤ δω < Rmax ,

(6.12)

where Rmax defines the distance to the nearest exact resonance. Remark 6.1. If exact resonances are absent, only approximate interactions occur and R = 0, while Rmax is defined as the border end of the inertial range of the wavenumbers. Otherwise, approximate interactions take place in rings with the widths (Rmax − R), i.e. resonance width defining approximate interactions is bounded from above. Notice that it is a nontrivial task to find the constant c appearing in the expression (6.9), corresponding to a specific dispersion function ω. However, the magnitude of R can easily be found numerically for given dispersion function and resonance conditions.

150

Wave turbulent regimes

Example 26: Radii R for gravity and capillary water waves Radius R has been found in numerical simulations [229, 230], both for capillary and surface gravity waves (the maximal spectral domain studied was 0 < m ≤ 2047, 0 < n ≤ 1023 and only scale-resonances were regarded). In the case of gravity surface waves, Rgrav = 10−5 and the diminishing of δω from 10−5 to 10−10 does not changes the number N of quasi-resonances. On the other hand, increasing δω from 10−4 to 10−3 we obtain 9N approximate interactions. For irrotational capillary waves, Rcap = 0, while they do not possess exact resonances. For both computations, the same discretization δ has been used. 6.3 Model of laminated turbulence The main statement of the model of laminated turbulence [112] is that any wave turbulent system consists of two layers: discrete and continuous. Main distinctions between these two layers are the following. Initial conditions The discrete layer is formed by exact and quasi-resonances governed by (6.10) and (6.11) and its time evolution depends on the initial conditions as explained in Chapter 4. In particular, dynamical phases are coherent, and can be used in real physical systems for controlling amplitude magnitudes without changing the total energy of the system. Another important point discussed in Section 2.2 is that the change of the form of the interaction domain – say, the ratio of the sides in a rectangular domain – can substantially diminish the number of possible resonances (see also [107, 127]). On the other hand, a continuous layer is formed by the approximate interactions defined by (6.12). In this case, provided that some additional assumptions hold (random phase approximation, locality of interactions, existence of an inertial interval, etc.), wave kinetic equations can be written out with KZ-spectra of energy. In the kinetic regime, time evolution of the wavefield does not depend either on the details of the initial conditions (such as initial energy distribution) or even on the magnitude of the initial energy – only the fact that it is conserved in the inertial interval is important. Interaction scales The assumption of locality of interactions (i.e. that only wavevectors with wave lengths with k of the same order interact) is necessary for deriving the wave kinetic equation. But it does not hold for exact resonances. Below we illustrate this statement with a few examples studied in more detail in [102]. In all the examples

6.3 Model of laminated turbulence

151

below, we denote as k− , k0 , k+ the minimal, intermediate, and maximum wavevectors kj , j = 1, 2, 3, respectively; the same indices are used for components of wavevectors. Example 27: Interaction domain for an atmospheric planetary wave The dispersion function in this case reads ω ∼ m/[n(n + 1)] and resonance conditions are given by (2.94). One of the resonance conditions is the triangle inequality, which allows us to conclude immediately that n+ < n0 + n− < 2n0 ,

(6.13)

while the fact that mj ≤ nj together with the explicit form of the frequency resonance condition yields = n− > and

= n− >

n0 +

5 1 − 4 2

5 1 1 n+ + − 2 4 2

for fixed n0 ,

for fixed n+ .

(6.14)

(6.15)

Finally, for fixed n− , we get n0 < n2− + n− − 1,



n+ < 2n0 < 2 n2− + n− − 1 .

(6.16)

We conclude finally that a wave with wavenumber n˜ can form a resonance only with waves whose wavenumbers belong to the interval #>=

? $ 5 1 1 2 n˜ + − , 2(n˜ + n˜ − 1) , 2 4 2

(6.17)

where & ' denotes the upper nearest integer. Thus, we have shown that waves with wavenumbers of orders n and n2 can form a resonance in this case. Example 28: Interaction domain for an ocean planetary wave The dispersion function now has the form ω ∼ 1/k with k = |k|, and the condition of a three-wave resonance can be rewritten as k+ =

k− k0 . k0 − k−

(6.18)

152

Wave turbulent regimes

Let us fix k− , then the maximum of (6.18) is reached if k0 = k− + 1, i.e. k+ ≤ k− (k− + 1).

(6.19)

Similar estimates can be obtained when either of k+ or k0 is fixed, i.e. waves with wavelengths of order k and of k 2 can form a resonance. Example 29: Interaction domain for a capillary-gravity water wave √ For dispersion function ω ∼ ( αk + k 3 ) with constant α, the conditions of threewave resonance yield 8 3 k+ < 2k0 , k0 < k− (6.20) 9 and correspondingly 1 ≤ k− ≤ k0 < k+ < 2k0
0, resonance clusters of two to six triads appear for ε = 10−7 , while for ε = 10−4 the overall number of resonances is about 106 , and more than 90% of all resonant triads coincide for σ = 74 and σ = 75. More results for irrotational gravity-capillary waves in water can be found in [121]; examples of quasi-resonant triads are given in [109] also for glycerine, benzol, etc.

174

Wave turbulent regimes

In the case of one-dimensional wavevector, k = k, the resonance conditions ω1 + ω2 = ω3 ,

k1 + k2 = k3

(6.97)

are satisfied if σ k12 (1 + r 2 ) + (1 + r)(1 + 7r + r 2 )1/2 = , 2g r(9 + 14r + 9r 2 )

(6.98)

where r = k2 /k1 . This condition was first obtained in [167]; the corresponding coupling coefficient is given in the Table 1.1. Recent laboratory experiments with irrotational gravity-capillary waves in mercury can be found in [70]. The probability density functions of the turbulent wave height are asymmetric and thus non-Gaussian. In the capillary region, the exponent is in fair agreement with weak turbulence theory while in the gravity region it depends on the experimental setting (forcing parameters). The surface wave height displays power-law spectra in both regimes with the scaling spectra being in disagreement with weak turbulence theory. This is a manifestation of a discrete wave turbulent regime (attributed in the paper to the finite-size effects). • In this chapter, we presented results on discrete wave turbulent regimes in various physical systems. It was also shown that in a mesoscopic regime, both discrete and continuous layers of wave turbulence can co-exist. The most interesting question then is how to describe the transition between these two layers. As resonance clusters are not local in k-space, they can transfer energy to substantially bigger scales if the system’s forcing takes place in small scales k (as in Fig. 6.1) or to substantially smaller scales in the opposite case. This means that for time-scales corresponding to the exact resonances, only a discrete regime is observable. At the next time-scale, quasi-resonances do appear with the smallest possible resonance broadening (see Fig. 2, [106]), which in turn trigger approximate interactions – at some critical resonance width – yielding energy spreading all over the k-space. This suggests that the energy cascade due to the transition from the discrete to the kinetic regime resembles more a set of sand-pile avalanches than a steady flow in k-space. A sand-pile model of such a transition was first introduced in [180] for gravity water waves. Suggestion of random evolution in this case allowed estimation of the critical energy spectrum as Eω ∼ ω−6 , which is substantially steeper than the KZ-spectrum for this case, Eω ∼ ω−4 . The critical spectrum Eω ∼ ω−6 has been confirmed in laboratory experiments [57] for some levels of forcing, while, for smaller forcing, spectra steeper than −6 have been identified. This is due to the oversimplified assumption about the purely random character of evolution during the transition between discrete and continuous layers of wave turbulence. Improvement of the sand-pile model and generalizing it for other dispersive wave

6.6 Discrete regimes in various wave systems

175

systems is a fascinating and quite nontrivial task; some new results in this direction can be found in [181]. • Last but not least, simply listing the subjects omitted from this book would be a few pages long. Just to mention some of them: influence of forcing and dissipation on the dynamics of a resonance cluster [157], effects of the general external parameters of the resonance dynamics [81], wave interactions in shear flows [51], wave collapses [226], physical implications of the explosive instability [172], stochastic instability of multiple nonlinear oscillations due to Arnold’s diffusion [37], etc. In fact, one of the most important subjects – the theoretical analysis of the stability of dynamical systems appearing while describing the discrete wave turbulent regime – has not even been mentioned. Obviously, in a volume of this size, only a few issues can be adequately discussed. This refers also to the list of references, which would be substantially extended if room allowed. Our main goal underlying the choice of topics has been to present the nonlinear resonance analysis as a new branch of mathematics – with its own terminology, theoretical results, computational methods, and examples of possible application areas. However, ideal mathematical models are frequently not adequate for a real physical system – an example of a linear pendulum regarded in Chapter 5 shows how many physical parameters have to be taken into account in order to make even this simplest model work. The factors influencing a model of the ideal pendulum include the change of the gravitational acceleration due to the latitude, interrelation of the masses of the different parts of the physical pendulum (the bob, the wire, the ring, the cap, etc.), buoyancy of the bob due to Archimedes’s law, and many others (see Table 5.1). If all of them were included in a mathematical model, it would become untractable analytically, though we can adjust these parameters according to the conditions of a specific laboratory experiment. Keeping this in mind, we were more interested in the presentation of the results of laboratory experiments demonstrating occurrences of nonlinear resonances in discrete and mesoscopic wave turbulent regimes than in further theoretical study of corresponding dynamical systems. Our description of these regimes in terms of resonance clustering can only be regarded as a mathematical “skeleton” of a real physical phenomenon – in this same sense as the equations q˙ = p,

p˙ = −q

(6.99)

describe a physical pendulum putting into motion grandfather clocks, and the Fourier harmonic A exp i[kx − ωt] (6.100) is relevant for your favorite radio program.

176

Wave turbulent regimes

Those who are interested in adding some physical “meat” to this skeleton should begin with the compendious monograph of Alex Craik [51], where a coherent up-to-date presentation (state of the art 1985) of theory and experiment on waveinteraction phenomena is given, both in fluids at rest and in shear flows. Beginning with linear waves, the author proceeds with an introduction to nonlinear theory, detailed discussion of three- and four-wave resonances (both conservative and nonconservative), and studies relevant to evolution equations (nonlinear Schrödinger, Davey–Stewartson, and Korteweg–de Vries equations). Some of the dynamical systems appearing in previous chapters (for primary clusters in three- and four-resonance systems, and for the butterfly cluster) are studied in Craik’s monograph, including the case of nonzero resonance broadening. All theoretical conclusions are illustrated by the results of numerical and laboratory experiments, the list of references consists of more than 700 entries. This book is sine qua non for those interested in physical applications of nonlinear resonance analysis. From the historical point of view, it is interesting to notice that the notions, nowadays standard, such as the wave kinetic equation, energy power spectrum, and wave turbulence theory are not to be met in [51], first published in 1985, though derivation and properties of the kinetic equation for gravity waves (under the name Zakharov’s equation) are already there. Contemporary terminology was first introduced in the classical volume written by Zakharov, Lvov, and Falkovich [260], and published seven years later, in 1992. In this book, the very foundations of kinetic wave turbulence theory were laid. For more recent general developments in this area, we refer to [138, 181, 257, 258]. An important novel subject first posted in [110] – influence of discretization necessary while performing computer simulations on the validity of results obtained for wave turbulent systems with continuous spectra – is thoroughly discussed in [154, 229] and in [23] (for irrotational and rotational nonlinear wave systems, respectively). 6.7 Open problems Methods presented in this book allow us to compute the frequencies of resonantly interacting waves in various wave systems, to construct resonance clusters, and to derive the explicit form of the corresponding dynamical systems. Some of these dynamical systems turned out to be integrable, but in general we have to rely on numerical simulations in order to get some understanding of the time evolution of a resonance cluster. This puts into the foreground the problem of stability of these dynamical systems. This type of problem is dealt with in the frame of dynamical systems

6.7 Open problems

177

theory. It is focused not on finding precise solutions, but rather on checking the existence of steady states, fixed points, periodic points, and similar subjects. This is outside the scope of the present section; excellent presentations can be found in [17, 224]. Below we present briefly some known results concerning not arbitrary dynamical systems but those describing resonances. They are studied in the context of the modern KAM theory and the theory of normal forms (e.g. [8, 140, 141, 142, 153, 176, 179, 182, 201] and many others). The classical results of the KAM theory are formulated in terms of the measure of the set of (exact and quasi-) resonances, which are excluded from consideration [152]. Later results – the Nekhoroshev theorem – show, at the cost of significant weakening of the error estimation, that some quasi-resonances can indeed be included in the standard KAM approach [8, 182], for some initial conditions. The Nekhoroshev theorem can be formulated as follows. Theorem 14. If a Hamiltonian has the form H (I, ϕ) = h(I ) + εf (I, ϕ),

(6.101)

where I ∈ Rn , ϕ ∈ T n , h is steep enough, and f is analytic, then there exist positive constants a, b ε0 , I0 , t0 such that ∀ε < ε0 and the inequality |I (t) − I (0)| ≤ I0 εa

(6.102)

holds for arbitrary t such that |t| ≤ t0 exp

ε b 0

ε

.

(6.103)

In order to use this theorem for the study of dynamical systems we have, of course, first to reformulate it in the spectral form. This has been done in [82], where also applicability of the Nekhoroshev theorem to the analysis of numerical and experimental data has been established. This result led to stability analysis of various asteroid systems, for instance asteroid families Koronis and Veritas [83, 191]. In particular, dynamically distinct regimes have been distinguished in [83]: (1) apparently stable, well described by the KAM theory; (2) chaotic but exponentially stable covered by the Nekhoroshev theorem, and (3) unstable chaotic regimes in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system. An interesting study in this direction is presented in [132], where the Nekhoroshev theorem (in spectral form) is used to identify different dynamical

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regimes – regular and chaotic motions – in the asteroid Main Belt. For this purpose, the authors introduce an auxiliary function depending on the Keplerian orbital elements, compute its Fast Fourier transform, and use its spectrum in order to distinguish between Nekhoroshev and nonNekhoroshev dynamical regimes. A basic mechanism of instability in a dynamical system having more than two degrees of freedom, called Arnold diffusion, is thoroughly discussed in [37], with a lot of enlightening examples. Notice that this instability is absent in completely integrable systems, for instance in an N-star cluster with arbitrary finite N and suitable coupling coefficients (more examples are given in Section 4.4). The approach pursued in the modern theory of normal forms [211] differs from KAM theory in the philosophy of averaging. In KAM theory, averaging is made over all angles at once, although yielding substantial inaccuracy in the description of the solutions’ trajectories (orbits). On the other hand, the goal of the theory of normal forms is to describe a solution’s trajectory as accurately as possible, the price being a substantially more complicated averaging over some angles [211]. The general Hamiltonian system with n degrees of freedom H (p, q) = H2 + εH3 + ε 2 H4 + · · ·

(6.104)

is regarded then in a truncated form 0(p, q) = H2 + ε H 03 + ε 2 H 04 + · · · + εm−2 H 0m H

(6.105)

and for solutions of the original Hamiltonian system (6.104), expressed in the original variables, the following estimate holds: |F (p, q) − F (p(0), q(0))| = O(εm−1 t).

(6.106)

0 yields approximate integrability This means in particular that the integrability of H of H , i.e. a chaotic component in the flow of the original Hamiltonian “is limited by the given error estimates and must be a small-scale phenomenon on a long timescale” [237]. Each normal form has at least two first integrals (conservation laws), any additional conservation law appearing in a specific system weakens the estimate (6.106), [233]. Sometimes, the integrability of the dynamical system can be established based on its normal form [28]. For instance, in [29], there is a special case of the Euler–Poisson system describing the motion of a rigid body with a fixed point as an autonomous sixth-order ODE system with one parameter. The resonant normal form is computed at stationary points, with resonances 1:2 and 1:3, and conditions of integrability (i.e. the existence of an additional local conservation law) are checked.

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179

Fig. 6.5 NR-diagrams for the resonance clusters given by (6.107), (6.108), and (6.109) respectively

However, robust application of these theoretical methods and results to real-life problems cannot circumvent at least the three crucial problems formulated below. Problem 1. In systems with cubic Hamiltonian H3 , any solution of resonance conditions (1.123) belongs to one q-class, i.e. ratios ωj /ωi are positive or negative rational numbers which is necessary for constructing KAM-tori. Generally speaking, we can present a resonance cluster as a set of corresponding frequencies. However, this representation does not define a dynamical system uniquely; indeed, regard frequency resonance conditions ω1 + ω2 = ω3 , ω4 + ω5 = ω3 ;

(6.107)

ω1 + ω 2 = ω3 , ω3 + ω 4 = ω5 ;

(6.108)

ω2 + ω 3 = ω 1 , ω 3 + ω 5 = ω 2 , ω 5 + ω 7 = ω 3 ;

(6.109)

appearing in various wave systems and studied in Chapter 4 (the cluster described by (6.115) appears in the laboratory experiments discussed in Section 4.8). In all three cases, the frequency set is (ω1 , ω2 , ω3 , ω4 , ω5 ),

(6.110)

but the corresponding NR-diagrams – and dynamical systems accordingly – have different forms shown in Fig. 6.5. Resulting dynamical systems can even have different numbers of degrees of freedom, shown in Fig. 6.5 in the middle and right panels; on the other hand, two butterfly clusters (Fig. 6.5, left and middle panels) have the same number of degrees of freedom. This means that the frequency set (6.110) describes a huge class of dynamical systems, with different stability characteristics. How to distinguish between these systems using their frequency sets? Problem 2. In systems with a quadric Hamiltonian H4 and with higher order Hamiltonians H5 , H6 , etc., a solution of the resonance conditions can be formed by wavevectors belonging to two or more different q-classes, as was shown in Chapter 2. Indeed, regard resonance conditions in a four-wave system of gravity water waves (1.124) with dispersion function ω ∼ (m2 + n2 )1/4 . The quartet (−15, −20), (15, 20), (−7, −24), (7, 24)

(6.111)

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is a solution of the four-wave resonance conditions (1.124) and a simple check (152 + 202 = 625 = 54 and 72 + 242 = 625 = 54 ) shows that all four wavevectors in (6.111) belong to the q-class Cl1 , with ωi /ωj = 1 for all i, j = 1, 2, 3, 4. On the other hand, the quartet (1, 1), (2, −1), (1, −1), (2, 1),

(6.112)

also giving a solution to (1.124), is formed by vectors from two different q-classes: Cl2 and Cl5 . This solution is the particular solution given by the parametric series (2.62) with t = 2; the series can be used for constructing other two-class resonances. This means in particular that for the quartet (6.112)  ω1 /ω3 = ω2 /ω4 = 1, but ω1 /ω2 = ω3 /ω4 = 4 2/5, (6.113) which implies that two-class resonances cannot be described by KAM-tori. The same applies for n-class resonances with arbitrary n > 1. How to include at least two-class resonances into stability analysis? Problem 3. Last but not least, as was discussed in Chapter 1, the definitions of resonance conditions in mathematics and in physics are different. Mathematical definition includes one resonance condition for frequencies ω1 ± · · · ± ωs = 0,

(6.114)

while the physical definition includes two conditions – for frequencies and wavevectors: ω1 ± · · · ± ωs = 0, k1 ± · · · ± ks = 0. (6.115) To show how overwhelming the contrast might be, let us regard an example – a three-wave system of ocean planetary waves with two space-variables, ω ∼ (m2 + n2 )−1/2 . Solutions to (6.115) for this case have been studied in Chapter 3 and the corresponding topological structure of the solution set is shown in Fig. 3.2 for m, n ≤ 50. Now regard solutions to (6.114) for the same ω ∼ (m2 + n2 )−1/2 . The q-class decomposition allows us to reduce (6.114) to 1 1 1 + = γ1 γ2 γ3

with integers γi , i = 1, 2, 3,

(6.116)

and (6.122) has a parametric solution γ1 = t (t + 1), γ2 = t + 1, γ3 = t,

t ∈ N,

(6.117)

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181

which implies that each q-class is represented by a connected graph (t = 1 is connected with t + 1 = 2; t = 2 is connected with t + 1 = 3, and so on). Coming back to the initial variables – wavevectors’ components mi , ni – we can prove also that the topological structure of the solution set of (6.114) is a connected graph (see [110, Section 1.2.4]). This means that the solutions to (6.20) are described by infinite-dimensional dynamical systems; their stability analysis will give us no information about finite resonance clusters appearing in the solutions to (6.115). Even in the finite spectral domain, the difference between the resonance structures of (6.114) and (6.115) is huge. Indeed, in the spectral domain m, n ≤ 50 solutions of (6.115) belonging to q-classes with index q ≤ 300 are described by 28 dynamical systems altogether, among them 18 systems contain only three complex variables and are integrable, while the largest dynamical system contains 12 complex variables. On the other hand, there exist 57 q-classes (see [127], Section 4.1) in this case and each class contains some solutions of (6.114). Each class is described by one dynamical system, i.e. we have altogether 57 systems, not 28 as above. Moreover, a minimal dynamical system now is not a triad but a cluster of six connected triads, while the largest dynamical system is formed by 186 triads. How to include wavevectors into mathematical analysis of resonances?

7 Epilogue

And why did you stir up the people in the bazaar, you vagrant, talking about the truth, of which you have no notion? M. Bulgakov Master and Margarita

The problem of small divisors, or commensurabilities of mean motions, has been the concern of great scientists for centuries [58]. Indeed, Kepler (1571–1630) described the motion of the planets as elliptic orbits not depending on time. This is an integrable Hamiltonian system which does not incorporate interactions between the planets; these interactions are of a few orders less than the interaction of the Sun with the planets and can be regarded as a small term εH0 in Hamiltonian representation H = Hint + εH0 . (7.1) Nevertheless, they should be taken into account if we would like to know, for instance, whether some day this small term might yield a collision of the Earth and the Sun. Known in celestial mechanics as the n-body problem, it was solved geometrically for the case n = 2 by Newton in 1687 [185], while a detailed analytical solution was provided by Euler in 1744 [68]. Lagrange analyzed the three-body problem and managed to reduce the initial system of 18 ODEs to seven and implicitly even to six equations [146]. Some particular solutions have been found, but by the middle of nineteenth century it was clear that a closed general solution was unlikely to be found, while secular terms could not be eliminated. Trying to work out the problem, Poincaré was the first to discover chaos in a dynamical system, the discovery estimated by Weirstrass as “epoch-making” [11]. In Poincaré’s formulation, “difficulties encountered in celestial mechanics on account of the existence of small divisors and approximate commensurabilities of mean motions are connected with the very nature of things and cannot be avoided” ([202]). In 182

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183

the 1950–60s the brilliant results of Kolmogorov, Arnold, and Moser on the existence of quasi-periodic solutions of Hamiltonian systems laid the foundations for the renowned KAM theory [7, 135, 174]. Its main statement concerning the small divisor problem can be formulated as follows. The set of nonresonant tori has a positive Liouville measure, while resonant and quasi-resonant tori form a set of zero measures, which is omitted from consideration. However, in real life, occurrence of resonance might cause dangerous consequences and therefore resonances have been studied extensively in various application areas. Methods presented in this book allow us to study nonlinear resonances independently of application area. Though not giving all the immediate answers, these methods help us to gain more knowledge about physical systems possessing nonlinear resonances. For instance, the use of q-class decomposition yields important information about the dynamics of the resonance system, without solving the corresponding equations of motion and even before the complete resonance set is constructed. Indeed, without even looking at the solutions we can conclude that wavevectors (1, 2) and (1, 8) cannot√form resonance oceanic planetary waves with dispersion function ω(m, n) ∼ 1/ m2 + n2 while belonging to different q-classes. This is important to know when planning laboratory experiments. Geometrical representation of a resonance cluster in the form of an NR-diagram is a simple and powerful tool for describing the general dynamical characteristics of clusters, again without solving the dynamical systems (size of the clusters, cluster reductions, angle- and scale-resonances, mixed cascade, integrability of corresponding dynamical systems, etc.). Numerically, resonance clustering can be used to develop NR-reduced models, diminishing the complexity of simulations by a few orders in big computational domains. Some recurrent patterns, found in nature, laboratory experiments, and numerical simulations can be interpreted based on resonance clustering. Let us summarize briefly the steps necessary when applying the theoretical findings on nonlinear resonances to a specific problem. • Regard the nonlinear partial differential equation with constant coefficients of arbitrary

order and number of variables; divide the variables into time- and space-variables; fix the boundary conditions. • Compute the eigenfunction of the linear part of the chosen equation and dispersion function according to the choice of boundary conditions. • Introduce a small parameter and use any perturbation method to construct the minimal dynamical system corresponding to a three-wave resonance: an example of this procedure is given in Chapter 1. If the resulting coupling coefficient is identically zero, apply the perturbation method to look for a four-wave resonance, and so on.

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Epilogue

• Provided the minimum order of resonance is established for which the coupling coeffi-

cient is not identically zero, write out the resonance conditions and solve them using the methods presented in Chapter 2. • Construct the corresponding resonance clustering, NR-diagram, and set of dynamical systems, applying the methods discussed in Chapter 3. • Use the results of Chapter 4 to establish which dynamical systems from the set obtained in the previous step are integrable; solve other systems numerically. • Utilize examples and case studies from Chapters 5 and 6 as prototypes to apply the results obtained to other real-life application areas.

Those who are not interested in the mathematical details of the computations can find an eigenfunction and corresponding dispersion function in the textbooks on partial differential equations. The explicit form of the resonance conditions and expressions for the coupling coefficient for a given evolutionary nonlinear dispersive PDE can be found in textbooks on specific application areas. The Glossary, together with the tables from Chapters 1–4, can then be used as a minimal handbook clarifying the definitions, notations, and results available. Computer programs given in the Appendix can be used either directly if appropriate or as prototypes for analogous programs with different dispersion functions and resonance conditions. A knowledge of resonance clustering and cluster dynamics gives rise to a number of novel questions which are presently open. How can you construct clustering for a transcendental dispersion function? Do any other integrable clusters exist besides those discussed in this book? Is it feasible to develop a general (perhaps approximate) description of resonance cluster dynamics based on a Kuramoto-like model for a system of connected nonlinear oscillators that are integrable primary clusters? How do you describe transition between resonance cluster dynamics and the rest of the spectra in wave turbulent systems? When is this transition important (observable) and when can it be neglected? How do you describe quantitatively the interrelation between KAM theory and wave turbulence theory? Keeping in mind that wave turbulent systems occur in medicine, biology, sociology, chemistry, etc., we can see how crucial this last problem is. Each new application will pose plenty of other problems to be attended to. Many new insights and fascinating findings still await discovery.

Appendix: Software Wolfgang Schreiner

The purpose of this appendix is to highlight the computational aspects of the nonlinear resonance analysis presented in this book. As an analogy, if we compare this theory with the theory of gas expansion, we now show how to build a car engine that any driver can use without being aware of how the engine is underpinned by theory. We start in Section A.1 by translating some of the previously described algorithmic methods to computer programs. In Section A.2, we demonstrate how to provide access to such programs via web-based service interfaces. In Section A.3, we sketch the integration of such services into a virtual laboratory for supporting research and education. An electronic supplement to this chapter with free access to the described software and services is available at http://www.risc.uni-linz.ac.at/people/schreine/Wave

A.1 Mathematical software We start this section by discussing how to efficiently solve by computer programs some of the resonance conditions presented in Chapter 2. Three-wave resonances In Section 2.2, the dispersion function for ocean planetary waves was presented as ω ∼ (m2 + n2 )−1/2 . We now address the problem of solving the corresponding resonance condition for three waves. In more detail, given a domain bound D ∈ N,

185

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Software (Wolfgang Schreiner)

we want to compute the set of all wavevectors (m1 , n1 ), (m2 , n2 ), (m3 , n3 ) ∈ Z2 that satisfy the following conditions: m21 + n21 ≤ D 2 ∧ m22 + n22 ≤ D 2 ∧ m23 + n23 " " ≤ D 2 ∧ 1/ m21 + n21 + 1/ m22 + n22 " = 1/ m23 + n23 ∧ n1 ± n2 = n3 . This formulation would suggest an (hopelessly inefficient) algorithm that constructs all (2D + 1)6 possible combinations of values for m1 , n1 , m2 , n2 , m3 , n3 and checks whether they satisfy the above criterion. However, as discussed in Section 2.2, by applying results from number theory, the conditions can be reformulated as follows: ∃q ∈ N : ∀p ∈ N : (p is prime ∧ p|q ⇒ p mod 4 = 3 ∧ p2 | q) ∧ ∃γ1 , γ2 , γ3 ∈ N : 1/γ1 + 1/γ2 = 1/γ3 ∧ γ12 · q ≤ D 2 ∧ γ22 · q ≤ D 2 ∧ γ32 · q ≤ D 2 ∧ m21 + n21 = γ12 · q ∧ m22 + n22 = γ22 · q ∧ m23 + n23 = γ32 · q ∧ n1 ± n2 = n3 . This formulation suggests an algorithm where certain combinations of q-class indices and wavevector weights γ1 , γ2 , γ3 are investigated: we consider only • those values for q that cannot be divided by the square of any prime number and have no

prime factor of the form 4u + 3, • those combinations of γ1 , γ2 , γ3 that satisfy 1/γ1 + 1/γ2 = 1/γ3 , • those combinations of q, γ1 , γ2 , γ3 that satisfy γ12 ·q ≤ D 2 , γ22 ·q ≤ D 2 , and γ32 ·q ≤ D 2 .

Since comparatively few such combinations of q and γi , i = 1, 2, 3 exist, the “search space” of the algorithm is considerably reduced compared with the naive 2 version. For each combination, we decompose a sum of squares, i.e. we 2 γ2i · q into 2 find all wavevectors ki = (mi , ni ) with mi + ni = γi · q. For all possible combinations of k1 , k2 , k3 , we check whether the condition n1 ± n2 = n3 is satisfied. If yes, we have found a solution. The whole algorithm is depicted as Algorithm 1.

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Algorithm 1 Solving resonance conditions for three waves function S OLVE 3W(D) R←∅ for all q ≤ D 2 do if ∀p ∈ N : (p is prime ∧ p|q ⇒ p mod 4 = 3 ∧ p2 | q) then for all γ1 , γ2 , γ3 ≤ D do if 1/γ1 + 1/γ2 = 1/γ3 then if γ12 · q ≤ D 2 ∧ γ22 · q ≤ D 2 ∧ γ32 · q ≤ D 2 then w1s ← D ECOMPOSE(γ12 · q) w2s ← D ECOMPOSE(γ22 · q) w3s ← D ECOMPOSE(γ32 · q) for all 0m1 , n1 1 ∈ w1s, 0m2 , n2 1 ∈ w2s, 0m3 , n3 1 ∈ w3s do if n1 ± n2 = n3 then R ← R ∪ {00m1 , n1 1, 0m2 , n2 1, 0m3 , n3 11} end if end for end if end if end for end if end for return R end function

The problem can be further simplified by considering only solutions in (m1 , n1 ), (m2 , n2 ), (m3 , n3 ) ∈ N2 . From these, all solutions with negative components can be constructed as follows: • Since only m2i appears in the resonance condition, the signs of all mi may be (indepen-

dently from each other) flipped to yield corresponding solutions with negative mi . • For each solution with components 0n1 , n2 , n3 1, the following transformations may be performed to yield solutions with negative components: 0−n1 , −n2 , −n3 1, 0n1 , −n2 , n3 1, 0−n1 , n2 , −n3 1.

We can easily check that by these transformations each solution form with one or more negative components can be derived from a solution with only positive components (possibly by switching vector indices 1 and 2). Since the construction is straightforward and not very interesting, it is subsequently neglected. The actual implementation of the algorithm proceeds as follows: (i) (ii) (iii) (iv)

A table primes of all prime numbers up to D is computed. Using primes, the table qs of all legal q ≤ D 2 is constructed. The table gammas of all legal combinations γ1 , γ2 , γ3 ≤ D is constructed. From qs and gammas, the legal combinations are constructed, the decompositions into sums of square roots are performed, and the solutions are constructed as described above.

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Software (Wolfgang Schreiner)

An initial prototype implementation of the algorithm in Mathematica is described in [127]. This prototype implementation took (partially due to an ineffective programming style) a few minutes to compute the solutions for D = 100. A more efficient form of the implementation [35] is shown below: Generateq[n_Integer,X]/; (* generate qs *) SquareFreeQ[n]&&!MemberQ[Mod[PrimeFactorList[n],4],3]:= n; Generateq[n_Integer,X]:=Sequence[]; Generateq[upperbound_Integer]:= Table[Generateq[q,X],{q,1,upperbound}]; GenerateGamma[Gamma1_Integer,Gamma2_Integer]/;(* generate gammas *) Mod[Gamma1 Gamma2,Gamma1+Gamma2]\[Equal]0:= Sequence[{Gamma2,Gamma1,Gamma1 Gamma2/(Gamma1+Gamma2)}, {Gamma1, Gamma2,Gamma1 Gamma2/(Gamma1+Gamma2)}]; GenerateGamma[Gamma1_Integer,Gamma2_Integer]:=Sequence[]; GenerateGamma[upperbound_]:=Flatten[{ Table[GenerateGamma[Gamma1,Gamma2], {Gamma1,1,upperbound},{Gamma2,1,Gamma1-1}], {Table[{Gamma, Gamma,Gamma/2},{Gamma,2,upperbound,2}]}}, 2] CreateSolutionSet[VectorLength,D_Integer]:= Module[{Q,G,SumOf2Squares,CreateMN,CheckSolution}, SumOf2Squares[i_Integer]:= SumOf2Squares[i]= Select[SumOfSquaresRepresentations[2,i],(0<Min[#])&]; SumOf2Squares[l:{___Integer}]:=SumOf2Squares/@l; CreateMN[q_Integer,Gamma:{__Integer}]:= Outer[Global‘Solution,Sequence@@SumOf2Squares[q Gamma^2],1]; CreateMN[q_Integer]:= CreateMN[q,#]&/@Select[G,(Max[#]\[LessEqual]D/Sqrt[q])&]; CheckSolution[ Global‘Solution[{m1_Integer,n1_Integer},{m2_Integer, n2_Integer},{m3_Integer,n3_Integer}]]/; n1-n2==n3||n1+n2==n3:=True; CheckSolution[___]:=False; Q=Generateq[D^2]; G=GenerateGamma[D]; sol=Flatten[CreateMN/@Q]; Select[sol,CheckSolution] ];

(* solve condition *)

The last four lines of the source show the core computation performed: generating the table of q values, generating the table of γi , constructing the raw solutions by combining the wavevectors derived from the decompositions into sums of squares, and finally discarding those that do not pass the linearity check.

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This program has the advantage that it is very short, because it makes use of Mathematica’s mathematical facilities, namely the procedures SquareFreeQ (square-free test), PrimeFactorList (prime number factorization), SumOfSquaresRepresentations (decomposition into sum of squares),

as well of Mathematica’s rule-oriented programming style and its notation Table [expr(i), {i, min, max}]) for building tables of values expri with i ∈ [min, max]. This Mathematica program solves the case D = 100 in about half a minute. However, it takes many minutes for cases D ≥ 300; here its applicability apparently reaches its limit. As an attempt at a more efficient solution, a new implementation in Java is presented below [35]. Its core function is (compare with the last lines of the Mathematica code above): S o l u t i o n S e t Solve3W ( i n t D) { P r i m e T a b l e p r i m e s = new P r i m e T a b l e (D ) ; QTable q s = new QTable (D∗D, p r i m e s ) ; GammaTable gammas = new GammaTable (D ) ; R o o t T a b l e r o o t s = new R o o t T a b l e (D∗D ) ; r e t u r n new S o l u t i o n S e t ( qs , gammas , r o o t s , D ) ; }

The function for the selection of valid combinations of q, γ1 , γ2 , γ3 is: S o l u t i o n S e t ( Qs qs , GammaTable gammas , R o o t T a b l e r o o t s , i n t D) { l o n g bound = D∗D; i n t qn = q s . l e n g t h ( ) ; f o r ( i n t i = 0 ; i bound ) c o n t i n u e ; l o n g g3 = gammas . gamma3 ( j ) ; l o n g v3 = q∗ g3 ∗ g3 ; i f ( v3 > bound ) c o n t i n u e ; I n t T a b l e P a i r p1 = decompose ( ( i n t ) v1 , r o o t s ) ; i f ( p1 == n u l l ) c o n t i n u e ; I n t T a b l e P a i r p2 = decompose ( ( i n t ) v2 , r o o t s ) ; i f ( p2 == n u l l ) c o n t i n u e ;

190

Software (Wolfgang Schreiner) I n t T a b l e P a i r p3 = decompose ( ( i n t ) v3 , r o o t s ) ; i f ( p3 == n u l l ) c o n t i n u e ; combine ( p1 , p2 , p3 , s o l s ) ; }

} }

As we can see, bound checks are performed as early as possible in order to find invalid combinations as soon as possible, in particular before any attempt at a decomposition into a sum of squares is made. Different from the Mathematica solution, the Java program implements all mathematical functionality from scratch, in particular: • Integer square root computation: the program builds a table roots of all square numbers

up to D 2 ; to compute the integer square root isqrt(v) of some integer v ≤ D 2 , we use a binary search to find the index i with roots[i] ≤ v < roots[i + 1] and return it as the result (at most log D steps are required). • Decomposition into sums of squares: to decompose some integer v ≤ D 2 into all possible √ sums of squares, we apply a naive algorithm which computes, for every m ≤ v, n := isqrt(v − m2 ); if m2 + n2 = v, the pair 0m, n1 is accepted as a solution.

The complete source code of the program is about 500 lines long (about ten times the size of the Mathematica program). The program solves the D = 100 case in less than half a second, so it is at least 60 times faster. A more detailed performance evaluation is given below.

Four-wave resonances In Section 2.2, the dispersion function for gravity water waves was presented as ω ∼ (m2 + n2 )−1/4 . We now address the problem of solving the corresponding resonance condition for four waves. In more detail, given a domain bound D ∈ N, we want to compute the set of all wavevectors (m1 , n1 ), (m2 , n2 ), (m3 , n3 ), (m4 , n4 ) ∈ Z2 that satisfy the following conditions: m21 + n21 ≤ D 2 ∧ m22 + n22 ≤ D 2 ∧ m23 + n23 ≤ D 2 ∧ m24 + n24 ≤ D 2 ∧ " " " " 4 m21 + n21 + 4 m22 + n22 = 4 m23 + n23 + 4 m24 + n24 ∧ m1 + m2 = m3 + m4 ∧ n1 + n2 = n3 + n4 . As discussed in Section 2.2, by applying results from number theory, the condition can be reformulated as follows:

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∃q ∈ N : ∀p ∈ N : (p is prime ∧ p|q ⇒ (p mod 4 = 3 ⇒ p 2 |q ∧ p3 |q) ∧ p4 | q) ∧ ∃γ1 , γ2 , γ3 , γ4 ∈ N : γ1 + γ2 = γ3 + γ4 ∧ γ14 · q ≤ D 2 ∧ γ24 · q ≤ D 2 ∧ γ34 · q ≤ D 2 ∧ γ44 · q ≤ D 2 ∧ m21 + n21 = γ14 · q ∧ m22 + n22 = γ24 · q ∧ m23 + n23 = γ34 · q ∧ m24 + n24 = γ44 · q ∧ m1 + m2 = m3 + m4 ∧ n1 + n2 = n3 + n4 .

In analogy to the previous three-wave case, this formulation suggests an algorithm where certain combinations of wavevector weights γ1 ,γ2 ,γ3 ,γ4 and q-class indices are investigated. We consider only: • those values for q that cannot be divided by the fourth power of any prime number and

where all prime factors of form 4u + 3 have degree 2, • those combinations of γ1 , γ2 , γ3 , γ4 that satisfy γ1 + γ2 = γ3 + γ4 , • those combinations of q, γ1 , γ2 , γ3 , γ4 that satisfy γ14 ·q ≤ D 2 , γ24 ·q ≤ D 2 , γ34 ·q ≤ D 2 ,

and γ44 · q ≤ D 2 .

For each such combination of q and γi , i = 1, 2, 3, 4, we decompose γi4 · q into a sum of squares; for all possible combinations of the results, we check whether the condition m1 + m2 = m3 + m4 ∧ n1 + n2 = n3 + n4 is satisfied. If yes, we have found a solution. Actually, we are not much interested in those solutions that are collinear, i.e. where the set {|m1 /n1 |, |m2 /n2 |, |m3 /n3 |, |m4 /n4 |} has only one element; such solutions are not recorded in the solution set. Furthermore, we may differentiate between those solutions with at most two different wave vector (angle-resonances) and the 1∧ |{|m s o l u t i o n s = new HashSet < S t r i n g > ( ) ; C a n o n i c a l S o l u t i o n s o l u t i o n = new C a n o n i c a l S o l u t i o n ( ) ; i n t l 1 = p1 . l e n g t h ( ) ; i n t l 2 = p2 . l e n g t h ( ) ; i n t l 3 = p3 . l e n g t h ( ) ; i n t l 4 = p4 . l e n g t h ( ) ; f o r ( i n t i = 0 ; i < l 1 ; i ++) { i n t m1 = p1 .m. g e t ( i ) ; i n t n1 = p1 . n . g e t ( i ) ; f o r ( i n t j = 0 ; j < l 2 ; j ++) { i n t m2 = p2 .m. g e t ( j ) ; i n t n2 = p2 . n . g e t ( j ) ; f o r ( i n t k = 0 ; k< l 3 ; k ++) { i n t m3 = p3 .m. g e t ( k ) ; i n t n3 = p3 . n . g e t ( k ) ; f o r ( i n t l = 0 ; l < l 4 ; l ++) { i n t m4 = p4 .m. g e t ( l ) ; i n t n4 = p4 . n . g e t ( l ) ; i f ( m1+m2 ! = m3+m4 | | n1+n2 ! = n3+n4 ) c o n t i n u e ; i f ( c o l l i n e a r ( m1 , n1 , m2 , n2 , m3 , n3 , m4 , n4 ) ) { c o l l i n e a r = c o l l i n e a r +1; continue ; } i f ( ! a n g l e ( m1 , n1 , m2 , n2 , m3 , n3 , m4 , n4 ) ) continue ; b o o l e a n okay = s o l u t i o n . c a n o n i z e ( m1 , n1 , m2 , n2 , m3 , n3 , m4 , n4 ) ; i f ( ! okay ) c o n t i n u e ; i n t m10 = s o l . m1 ( ) ; i n t n10 = s o l . n1 ( ) ; i n t m20 = s o l . m2 ( ) ; i n t n20 = s o l . n2 ( ) ; i n t m30 = s o l . m3 ( ) ; i n t n30 = s o l . n3 ( ) ; i n t m40 = s o l . m4 ( ) ; i n t n40 = s o l . n4 ( ) ; S t r i n g s 1 = m10+ " , " +n10+ " , " +m20+ " , " +n20+ " , " + m30+ " , " +n30+ " , " +m40+ " , " +n40 ; i f ( s o l u t i o n s . c o n t a i n s ( s1 ) ) continue ; S t r i n g s 2 = n10+ " , " +m10+ " , " + n20+ " , " +m20+ " , " + n30+ " , " +m30+ " , " + n40+ " , " +m40 ; s o l u t i o n s . add ( s 1 ) ; s o l u t i o n s . add ( s 2 ) ; m1t . add ( m10 ) ; n 1 t . add ( n10 ) ; m2t . add ( m20 ) ; n 2 t . add ( n20 ) ; m3t . add ( m30 ) ; n 3 t . add ( n30 ) ; m4t . add ( m40 ) ; n 4 t . add ( n40 ) ; } } } } }

193

194

Software (Wolfgang Schreiner) Table A.1. Solver benchmarks

Type

D = 10

D = 20

D = 30

D = 50

3-Wave n1 + n2 = n3 n1 − n2 = n3 4-Wave (angle) 4-Wave (scale) Angle Scale Collinear

Function[ Evaluate[DeleteCases[vars,t][[i]]], Evaluate[linwav[[i]] /. (Rule@@#& /@ Transpose[ {params,paramvalues[[j]] }] )]], {i,n},{j,num}]] ]

A.2 Mathematical services The classical approach to let users execute mathematical software such as that presented in the previous section is to provide programs, in source or as executables, for download and local execution on the user’s computer. However, this raises a

A.2 Mathematical services

203

lot of problems since full applications typically consist of multiple components that are written in different languages, depend on various libraries, and/or need interpreters (e.g. computer algebra systems) for execution. We therefore follow a more modern trend in computer science, which turns away from stand-alone software that is installed on local computers and can be only executed locally via a graphical user interface, and proceeds towards serviceoriented software [78]. Here the software is installed on some server computer and an interface is developed such that the software can be executed on the server from any computer connected to the Internet via a conventional Web browser. Various projects in computer mathematics have pursued research on corresponding middleware for mathematical Web services, see for instance MONET (Mathematics on the Net) [173] or our own MathBroker project [10, 164]. Based on previous developments [127], we have deployed, using simple and generally available Web technologies such as PHP, a prototypic service that implements a typical workflow for investigating nonlinear resonances (see Figure A.1). This workflow consists of: • computing the solutions of a three-wave resonance condition, • plotting the topological structure of the solution set, • computing the coupling coefficient of the dynamical system corresponding to a solution,

and • visualizing the solution of the initial PDE by plotting the modes’ amplitudes and constructing an animation for a chosen small parameter 0 < ε  1.

To use our Web interface, we have to: • Enter in the small text field at the top of the browser window a bound D ∈ N; by

pressing the button “Create Solution Set,” the set S of all solution vectors w1 , w2 , w3 with |wi | ≤ D of the condition is computed and displayed (in a linear Mathematica representation) in the big text area below (the Java program presented in Section A.1). • Take this solution set (or provide another one) and press the button “Plot Topology”; now the topological structure of the solution set is computed as a list of hypergraphs which are displayed as images in the right frame of the browser window (the Mathematica program described in Section A.1). • Input one solution in the text field below the text area with the complete solution set and press “Compute Coefficients” to compute the coupling coefficients of the dynamical system corresponding to the chosen solution (the Mathematica program described in Section A.1). • Enter the initial values for “slow” amplitudes, phases, and small parameter ε, and press “Plot Amplitudes” to construct the 2D image of the amplitudes and the 3D-animation for the time-evolution, during one period, of the complete solution of the initial nonlinear PDE.

204

Software (Wolfgang Schreiner)

Fig. A.1 Web interface of a mathematical service

In the following, we explain the technical realization of the first two phases of the web interface (the other two proceed in a similar fashion). The first phase is depicted in the top diagram in Fig. A.2: by pressing the button “Create Solution Set”, (1) the Web browser creates a request to execute the PHP script CreateSolutionSet.php with argument D and sends it to the Web server which invokes the PHP engine to execute the script. The script (2) starts the program in the Java archive Wave3.jar with command line argument -s D. The program (3) prints the result S on its standard output stream from where it is captured by the script. Finally (4) the script creates a Web page embedding the result and returns it to the Web browser where it is displayed. The core of the corresponding PHP code is as follows:

A.2 Mathematical services Client Computer

205 CENREC Server

D (1) CreateSolutionSet.php/D Create Solution Set

Web Server/ PHP Engine

(4) ..S.. (2) D

(3) S

S java −server −jar Wave3 −s Plot Topology

Create Solution Set

(1) PlotTopology.php/S

Web Server/ PHP Engine

(5) (2) PlotTopology[S]

(4) N

S (6) GET image−1.png

Mathematica

Plot Topology (3) Export["image−1.png"]

Fig. A.2 Implementation of a mathematical service

The local variable $domain is set to the value D retrieved from the Web interface. Then the command $command is constructed and executed to start the Java

206

Software (Wolfgang Schreiner)

virtual machine to run the program in the Java archive Wave3.jar to compute S. The result is printed to the standard output stream from where it is captured in the variable $result. From this, the HTML code of the result document is constructed, which is ultimately delivered to the browser. The execution of the second phase of the service proceeds as follows (see the bottom diagram in Fig. A.2): by pressing the button “Plot Topology”, (1) the Web browser creates a request to execute the PHP script PlotTopology.php with argument S and sends it to the Web server, which (2) starts Mathematica to execute a command PlotTopology[...S...]. Mathematica (3) generates a couple of images as files on the disk and (4) prints their number N on its output stream from where it is retrieved. The script then (5) creates a Web page with links to the images and returns it to the Web browser; to display the page, the browser (6) issues to the Web server a sequence of requests for the images which are delivered correspondingly. While conceptually simple, the programming of such a service is a low-level and error-prone task; as described in the following subsection, we strive in the long term for a higher-level solution. A.3 A web portal As shown in the previous sections, the technology is available to turn theoretical results into efficiently executable services that may be easily accessed by users via the World Wide Web. Our ultimate goal is to provide a “virtual laboratory” for the areas where nonlinear resonances do occur that demonstrates the practical applicability of the theoretical results to students, researchers, and engineers. Such a laboratory may serve educational purposes by providing online training facilities, for use in graduate and in post-graduate education. Furthermore, it may provide a framework for researchers by allowing them to perform online virtual experiments without the necessity of having access to actual physical equipment. As a first step towards this goal, we have started development of the Web portal CENREC – Center for Nonlinear Resonances Computations, which is currently in alpha status. The portal, which is based on the free content management system Drupal, will ultimately provide the following major functionality: • a Wiki-based hypertext encyclopedia, which surveys the main subjects of the domain; • a bibliographic database, which presents the state of the art in the corresponding research; • a collection of services, which allow via Web interfaces the easy and location-

independent use of programs for solving problems in the field.

A.3 A web portal

207

Fig. A.3 An encyclopedia article

The encyclopedia is based on the free software package MediaWiki, which has been developed for the online encyclopedia Wikipedia. It allows us to write hypertext articles (including multimedia objects such as images or animations) that contain links to other articles and resources of the portal. Via a LATEX-like input language, it is possible to write mathematical formulas, which are rendered as corresponding images. A sample article is depicted in Fig. A.3. The bibliographic database is based on Drupal’s “Bibliography” module; it collects literature in a structured form for browsing and querying with respect to various search criteria. Every entry is equipped with the usual bibliographic information (author, title, publication data, URL in case of electronic availability, abstract, list of keywords), see Fig. A.4 for the visual display of a sample item. Every item can be electronically referenced via a URL, in particular by articles of the encyclopedia. As for the portal’s service framework, we envisage a solution along the lines of model-driven engineering (MDE) of software [216] where low-level (system/platform-dependent) code is automatically generated from a high-level (domain-oriented) model. In the case of Web-based services, the model can naturally be decomposed into two components, a workflow model, which represents

208

Software (Wolfgang Schreiner)

Fig. A.4 A bibliographic item

an abstract view of the actual computations performed on the server, and an interface model, which represents a (simplified) description of the Web page displayed in the client’s browser; some vertices of the workflow (corresponding to user interactions) are linked to corresponding elements of the interface. To turn a locally developed program into a Web-based service, the programmer provides a package which contains all the files needed for executing the basic computation (libraries to be loaded by the interpreter, data files, etc.) together with descriptions of the workflow and interface models. This package is uploaded to the server where the actual service code (e.g. in PHP) is generated, and the service is deployed. Consequently, every mathematical programmer can, without particular knowledge of Web programming, make research results (in the form of executable programs) accessible to a wide target audience.

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Index

conservation law, 90, 91, 96, 97, 108, 110, 112, 114, 115, 125, 132, 133, 178 cubic, 107 quadratic, 107 time-dependent, 96, 110 time-independent, 96, 110 constant of motion Manley–Rowe form, 90, 92–4, 109, 114, 115, 125 coupling coefficient, xv, 19–21, 23, 25–9, 45, 66–8, 75, 77, 114, 119, 121, 123, 127, 166, 170, 171, 174, 178, 184, 201, 203 canonical, 28, 29 numerical, 27 physical, 27 criterion of instability Hasselmann, 93, 95 Safdi–Segur, 121 Wilhelmsson–Stenflo–Engelmann, 120

k-space, xv, 20, 44, 145, 152, 153, 156, 174 q-class, 32, 33, 35–7, 39, 41, 43, 45, 46, 49–53, 77, 179–81, 183 decomposition, 31, 32, 45, 46, 57, 58, 63, 77, 173, 180, 183 deficiency set, 52–4 index, 37, 43, 46, 47, 50, 53, 186, 191 multiplicative properties, 38 multiplicity, 46, 47, 49, 51 structural properties, 37 AA-butterfly, 85, 106, 108 dynamical system, 108 AA-ray, 108, 142 dynamical system, 108 AP-butterfly, 85, 106 dynamical system, 114 Clipping method, 122, 125, 126 cluster of quartets V-reduction, xv, 95 cluster of triads PP-reduction, xv, 95 cluster’s reduction, xv, 95 computational complexity, 45, 46, 51, 59 computational routine coupling coefficient, 199 resonance conditions four waves, 190 three waves, 186 solution set topological structure of, 195 connection type cluster of quartets D-, xv, 86 E-, xv, 86, 88 V-, xv, 86 cluster of triads AA, xv, 94, 110, 127 AP, xv, 94, 110, 127, 128 PP, xv, 94, 95, 110, 127, 128

degenerate initial conditions, 96, 99, 101 diagram NR-, xv, 11, 66, 76, 82, 83, 85, 86, 88, 89, 95, 109, 128, 169, 170, 179, 184 Feynman, 88 Wyld, 88 dispersion function, 12–14, 20, 22, 25–7, 57, 58, 64–6, 190 general form, 60 irrational, 32, 39, 57, 63 capillary waves, 41 gravity water waves, 43 internal waves, 39 ocean planetary waves, 35 sound waves, 41 rational, 57, 58, 63 trancendent, 63 dynamical invariant, 96–8, 100, 131–4, 136 dynamical phase, xv, 92, 97, 103–7, 110, 126, 128, 150, 171 quartet, 92, 117 triad, 91, 96, 110, 119

221

222 dynamical system, 28, 67, 68, 72–5, 88 generic, xv incidence matrix, 73 isomorphic, 67, 73 multigraph, 74 non-isomorphic, 67 primary, 25, 66 quartet, 92 triad, 90, 99 quartet, 24, 116, 120 resonance, 93 solution trajectory, 100 stability, 176, 178 triad, 23, 29, 97, 119 Fourier, 1–3 analysis, 1, 2, 4, 5, 7 harmonic, 12–16, 89, 175 series, 4 space, 20, 163 sum, 4 transform, 163, 178 transformation, 10, 20 Galerkin truncation, 122, 125, 126 Hamiltonian, 19–23, 90–2, 97, 103, 104, 106, 121, 125, 132, 133, 140, 162, 163, 165, 166, 177, 182 cubic, 21, 22, 123, 165, 179 dynamics, 20 equation, 20, 162, 163 expansion, 29 formalism, xii, 10, 19, 20 function, 21 quadratic, 22, 163 quadric, 21–3, 123, 179 system, 21, 126, 178 inertial interval, 144–6, 150, 152, 153, 155–7 instability Arnold diffusion, 178 Benjamin–Feir, 171, 172 decay, xv, 25, 93, 117 quartet, 94 triad, 93 explosive, 25, 117, 118, 121 criterion for a quartet, 118 criterion for a triad, 118 quartet, 120 triad, 118, 119 McLean, 171, 172 modulational, 171, 172 intra-seasonal oscillations, 103, 171 KAM-theory, 177, 178, 183, 184 KZ-spectra, 145, 150, 153, 174 gaps, 146, 153 Lagrangian, 141 dynamics, 19, 20 laminated wave turbulence, 146, 150, 153

Index multi-scale method, 10, 16 N-star dynamical system A-connection, 110 normal form, 9, 10, 115, 165, 177, 178 NR-reduced model, 122, 125, 126, 183 ODE, 5, 10, 90, 200 nonlinear, 8–10 parametric series, 44 atmospheric planetary waves one-parametric, 57 two-parametric, 57 gravity water waves five-parametric, 44 three-parametric, 44 tridents, 44 trivial quartets, 44 sound waves, 42 PDE, 4, 7, 10–13, 19, 29 classification mathematical, 11, 12 physical, 10, 12 dispersive, 12, 14, 16, 88 elliptic, 11 evolutionary, 12, 14, 16, 88 hyperbolic, 11 linear, 1, 12, 13 nondispersive, 14 nonlinear, 1, 7, 10, 200, 201, 203 parabolic, 11 stationary, 13 pendulum, ix, 1, 23, 129, 130, 136 elastic, 92, 115, 130, 138, 140–2 damped, 142 forced, 142 linear, 15, 19, 130, 138, 140, 143, 175 damped, 131, 133 physical, 137, 143, 175 Wilberforce, 142, 143 Poincaré, 1, 10, 182 section, 112, 140 transformation, 1, 9, 10 PP-butterfly, 85, 106, 107, 111, 112, 124 dynamical system, 106 quartet, xiv, 88 1-pair, xv, 86, 94 2-pair, xv, 86, 94 quasi-resonance, 155 resonance angle-, xv, 44, 76, 77, 81, 82, 86, 88, 191 broadening, 146, 147, 156, 174, 176 exact, xiv, 146, 149, 157, 174, 177 four-wave, 19, 24, 25, 64, 65, 76, 77, 92, 118, 190 nonlinear, 25, 88 quasi, 146, 149, 174, 177 scale-, xv, 76–8, 81, 82, 86, 88, 191

Index three-wave, 23, 25, 29, 32, 50, 64, 65, 69, 71, 76, 77, 118, 185, 199 resonance cluster, 64–8, 79, 88, 123, 179 butterfly, 169 connection type, 88 generic, xiv, 66, 142 hypergraph representation, 71–3, 75 mixed cascade, 76, 81, 82 multigraph representation, 74, 75 of triads butterfly, 106, 107, 110, 176, 179 N-star, 109, 110, 169, 178 ray, 107, 108 star, 110 primary, xiv, 25, 67, 88, 142, 176 four waves, 67 three waves, 67 quartet, 45, 64, 76–80 collinear, 77 common, 78–80 non-collinear, 77, 79–81 non-trident, 78, 82 trident, 77–80 size, xiv, 79 resonance conditions, 7, 16, 18, 21, 27, 30, 50, 68, 69, 76, 89, 142, 185, 201 s waves, 16, 118 four waves, xiv, 23, 43, 192, 194 in mathematics, 9, 23, 180 in physics, 16, 23, 180 three waves, xiv, 16, 23, 35, 39, 41, 45, 51, 57, 91, 118, 185, 187, 194 resonance curve, 64, 65 resonance set, xiv, 64–6, 79, 81 hypergraph presentation, 203 structure of geometrical, xiv, 65, 66, 76 topological, xiv, 66, 67, 82, 128, 180, 181 resonances density of, 38 small divisor problem, 10, 182 superposition principle, 4, 15 theorem Fermat’s last, 32, 41 of Besikovitch, 33, 35 of Euler, 36, 37, 41, 43, 47 of Gauss, 39 of Jacobi, 37 of Lindemann–Weierstrass, 63 of Nekhoroshev, 177 of Poincaré, 9, 10 of Thue–Siegel–Roth, 147 on density of equal norms, 37 on dimension of flows, 167 on partitioning, 61 on the sign of vorticity, 168 triad, xiv, 66, 88

223 A-mode, xv, 83, 85, 86, 93, 94, 101, 104–6, 109, 118, 129, 155, 157, 171 P-mode, xv, 83, 85, 93, 94, 101, 104–6, 109, 118, 129 variable action-angle, 19, 20, 133 amplitude-phase, xv, 92, 107, 109, 110 canonical, xv, 10, 19, 20, 25, 29, 90, 101, 110, 112, 116, 120, 123, 140, 162 physical, xv, 10, 19, 20, 26, 28, 29, 101, 103, 112, 114, 123, 125 space, 7, 12, 13, 19, 180, 183 time, 7, 12, 13, 183 fast, 15, 142 slow, 15, 16, 142 vorticity, 16, 25, 27, 29, 118, 123, 125, 152, 156, 161, 162, 164, 166–70, 172, 173, 200 wave kinetic equation, 145, 146, 150, 152, 176 wave turbulent regime discrete, 146, 153, 156, 157, 174, 175 energy cascade, 81, 146, 156, 158, 174 kinetic, 146, 153, 155, 157, 174 mesoscopic, 146, 153, 174, 175 waves atmospheric planetary, 13, 21, 26, 27, 57, 101, 103, 112, 113, 122, 126, 138, 148, 151, 156, 171 capillary Helium, 155 capillary water, 21, 150, 154, 156, 166 irrotational, 32, 41, 150, 156, 157 rotational, 152, 161, 164, 168, 170 capillary-gravity, 128, 152 irrotational, 173 mercury, 174 rotational, 172, 173 drift, 21 gravity water, 21, 29, 43, 45, 63, 76, 77, 81, 147, 150, 172, 174, 190 irrotational, 171, 172 rotational, 172 inertial water, 152 internal, 46, 63 Kelvin, 21 ocean planetary, 14, 21, 27, 29, 35, 39, 45, 50, 62, 64–6, 75, 126, 151, 180, 185 plasma electrostatic in cylinder, 29 electrostatic unmagnetized, 29 magnetized cold, 29 magnetized hot, 29 MHD, 29 sound, 30, 41, 42, 45, 51, 148 spin, 21 water, 122 wavevector deficiency, 52 multiplicity, 56, 60, 79–81 weight, 46, 52, 186, 191