Peyresq lectures on nonlinear phenomena : [lectures of two summer schools, the INLN Summer Schools on Nonlinear Phenomena, in June 1998 and June 1999]

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Peyresq Lectures on

Nonlinear Phenomena

This page is intentionally left blank

Peyresq Lectures on

Nonlinear Phenomena

Editors

Robin Kaiser James Montaldi Institut Non Lineaire de Nice, France

1 > World Scientific

Singapore »New Jersey * London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-ta-Publication DaU A catalogue record for this book is available from the British Library.

PEYRESQ LECTURES ON NONLINEAR PHENOMENA Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4315-4

Printed in Singapore by Uto-Print

Preface Nonlinear science is a very broad domain, with its feet in mathematics, physics, chemistry, biology, medicine as well as in less exact sciences such as economics and sociology. Nineteenth century science was mostly linear and the progress towards an understanding of the diverse behaviour of non linear systems is among the most important general scientific advances of the twentieth century. The lectures contained in this book took place at two summer schools, the INLN Summer Schools on Nonlinear Phenomena, in June 1998 and June 1999. The Institut Non Linâire de Nice (INLN) is a pluridisciplinary institute interested in many aspects of nonlinear science, and the principal purpose of this ongoing series of summer schools is to introduce doctoral students, either from the INLN or form other institutions, to a range of topics that are outside of their own domain of research. The eight courses represented by these lecture notes therefore cover a broad area, describing analytic, geometric and experimental approaches to subjects as diverse as wound-healing, turbulence, elasticity, classical mechanics, semi-classical quantum theory, water waves and trapping atoms. It is hoped that the publication of these notes will be useful to others in the field(s) of nonlinear science. We would like to take this opportunity, as organizers of the two sum mer schools, to thank the Fondation Nicolas-Claude Fabri de Peiresc, which hosts our stay in the beautiful village of Peyresq in the French Alps, and in particular the president Mady Smets, for providing a wonderfully relaxed at mosphere, allowing the participants and lecturers to interact easily both on scientific and personal levels. We would also like to thank the local Direction Rtgionale du CNRS, for partially funding these summer schools. Robin Kaiser James Montaldi Valbonne, 2000

V


Addresses of Contributors Basile Audoly Yves Pomeau Laboratoire de Physique Statistique de I'Ecole Normale Supirieure 24 Rue Lhomond, 75231 Paris Cedex 05, France. Dominique Delande Laboratoire Kastler-Brossel Tour 12, Etage 1, Universite Pierre et Marie Curie 4, place Jussieu, F-75252 Paris Cedex 05, France. Gerard Iooss Michel LeBellac Eric Lombardi Robin Kaiser James Montaldi Institut Non Lineaire de Nice UMR CNRS-UNSA 6618 1361 route des Lucioles F-06560 Valbonne, France Philip Miani Centre for Mathematical Biology Mathematical Institute 24-29 St. Giles' Oxford 0X1 3LB

VII


CONTENTS

Preface Addresses of Contributors

v vii

Elasticity and Geometry B. Audoly & Y. Pomeau 1. Introduction 2. Differential geometry of 2D manifolds 2.1. Developable surfaces 2.2. Geometry of the Poincare" half-plane 3. Thin plate elasticity 3.1. Euler-Lagrange functional 3.2. Scaling of the FvK equations 3.3. Geometry and the FvK equations 3.4. Thin shells: an example 4. Buckling in thin film delamination 4.1. The straight sided blister 4.2. Telephone cord delamination Quantum Chaos D. Delande 1. What is Quantum Chaos? 1.1. Classical chaos 1.2. Quantum dynamcis 1.3. Semiclassical dynamics 1.4. Physical situations of interest 1.5. A simple example: the hydrogen atom in a magnetic field 2. Time scales — Energy scales 3. Statistical properties of energy levels — random matrix theory 3.1. Level dynamics 3.2. Statistical analysis of the spectral fluctuations fluctuations 3.3. Regular regime ix

1 1 2 5 7 11 13 15 17 23 28 29 33 37 37 37 38 40 42 44 46 48 48 51 53

x

Contents

3.4. Chaotic regime — random matrix theory 3.5. Random matrix theory — continued 4. Semiclassical approximation 4.1. Regular systems — EBK/WKB quantization 4.2. Semiclassical propagator 4.3. Green's function 4.4. Trace formula 4.5. Convergence properties of the trace formula 4.6. An example: the Helium atom 5. Conclusion

56 58 60 60 63 65 66 69 71 72

The Water-wave Problem as a Spatial Dynamical System G. Iooss 1. Introduction 2. Formulation as a reversible dynamical system 2.1. Case of one layer with surface tension at thefreesurface 2.2. Case of two layers without surface tension 3. The linearized problem 4. Basic codimension one reversible normal forms 4.1. Case (i) 4.2. Case (ii) 4.3. Case (iii) 4.4. Case (iv) 5. Typical results for finite depth problems 6. Infinite depth case 6.1. Spectrum of the linearized problem 6.2. Normal forms in infinite dimensions 6.3. Typical results

77

Cold Atoms and Multiple Scattering R. Kaiser 1. Classical model of Doppler cooling 1.1. Internal motion: elastically bound electron 1.2. Radiation forces acting on the atom: "classical approach" 1.3. Resonant radiation pressure 1.4. Dipole force

77 77 77 79 81 83 84 85 86 88 89 90 90 91 91 95 95 96 99 103 105

Contents

xi

1.5. Doppler cooling 2. Interferences in multiple scattering 2.1. Scattering cross section of single atoms 2.2. Multiple scattering samples in atomic physics 2.3. Dwell time 2.4. Coherent backscattering of light 2.5. Strong localization of light in atom? 3. Conclusion

106 110 110 112 113 114 119 124

An Introduction to Zakharov Theory of Weak Turbulence M. L. Bellac 1. Introduction 2. Hamiltonian formalism for water waves 2.1. Fundamental equations 2.2. Hamilton's equations of motion 2.3. The pertubative expansion 3. The normal form of the Hamiltonian 3.1. Ho = sum of harmonic oscillators 3.2. Nonlinear terms: three wave interactions 3.3. Nonlinear terms: four wave interactions 3.4. Dimensional analysis and scaling laws 3.5. Miscellaneous remarks 4. Kinetic equations 4.1. Derivation of the kinetic equations 4.2. Conservation laws 5. Stationary spectra of weak turbulence 5.1. Dimensional estimates 5.2. Zakharov transformation 5.3. Examples andfinalremarks

127 127 131 131 133 135 137 137 138 142 144 145 146 147 150 152 152 155 157

Phenomena Beyond All Orders and Bifurcations of Reversible Homoclinic Connections near Higher Resonances 161 E. Lombardi 1. Introduction 161 1.1. Phenomena beyond all orders in dynamical systems 161

xii

2.

3.

4. 5.

Contents

1.2. A little toy model:fromphenomena beyond any algebraic order to oscillatory integrals Exponential tools for evaluating monofrequencyoscillatory integrals 2.1. Rough exponential upper bounds 2.2. Sharp exponential upper bounds 2.3. Exponential equivalent: general theory 2.4. Exponential equivalent: strategy for nonlinear differential equation Resonances of reversible vector fields 3.1. Definitions 3.2. Linear classification of reversible fixed points 3.3. Nomenclature The 02+ioo resonance The (i(D0)2icoi resonance 5.1. Exponential asymptotics of bi-oscillatory integrals 5.2. Strategy for non linear differential equations

Mathematical Modelling in the Life Sciences: Applications in Pattern Formation and Wound Healing P. K. Maini 1. Introduction 2. Models for pattern formation and morphogenesis 2.1. Chemical pre-pattern models 2.2. Cell movement models 2.3. Cell rearrangement models 2.4. Applications 2.5. Coupling pattern generators 2.6. Domain growth 2.7. Discussion 3. Models for wound healing 3.1. Corneal wound healing 3.2. Dermal healing 3.3. Discussion 4. Conclusions

164 171 171 173 175 179 182 182 183 183 188 193 196 198

201 201 202 202 207 209 210 215 217 219 220 220 223 230 230

Contents

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian System J. Montaldi 1. Introduction 1.1. Hamilton's equations 1.2. Examples 1.3. Symmetry 1.4. Central force problem 1.5. Lie group actions 2. Noether's theorem and the momentum map 2.1. Noether's theorem 2.2. Equivariance of the momentum map 2.3. Reduction 2.4. Singular reduction 2.5. Symplectic slice and the reduced space 3. Relative equilibria 4. Bifurcations of (relative) equilibria 4.1. One degree of freedom 4.2. Higher degrees of freedom 5. Geometric bifurcations 6. Examples 6.1. Point vortices on the sphere 6.2. Point vortices in the plane 6.3. molecules

xiii

239 239 239 241 243 243 246 247 249 251 255 256 257 257 261 262 263 264 266 266 272 275

ELASTICITY A N D GEOMETRY BASILE AUDOLY YVES POMEAU Laboraioire de Physique Statistique de VEcolt Normale Sup&rieure We outline the general principles of thin plate elasticity, by emphasizing their connection with the classical results of differential geometry. The relevant FvK equation, can be solved in some specific cases, even though they are strongly and definitely nonlinear. We present two types of solutions. The first one concerns the contact of a spherical shell on a flat plane at increasing pressing forces, the second one is about the buckling of a thin film under pressure on a flat substrate, where we explain the observed "telephone-cord" pattern of delamination.

1

Introduction

This paper follows from a set of lectures by the two authors given in the beau tiful setting of the Rencontres de Peyresq, in the high country, north of Nice in late Spring 1999. Those lectures were devoted to the exposition of some re cent results in thin plate elasticity. This venerable field of classical mechanics is witnessing a renewal of interest because in parts of the new attraction of physicists and applied mathematician for everything linking classical geome try and observations made in everyday life. Those lectures were focused first on the general principles of thin plate elasticity, emphasizing as much as we have could their connection with the beautiful results of classical differential geometry. Below, we present a rather detailed derivation of Gauss Theorema egregium, stating the condition under which two surfaces can be mapped on each other without changing the curvilinear distances. Although this is of ten presented as obvious, the connection between this Theorema egregium and the laws of elasticity of thin plates is not so simple. Hopefully, we make this clearer in our derivation of the equations of Foppl-von Karman for thin plates. Later, we use those equations to analyze two physical problems. The first one concerns the way a spherical shell deforms when pressed on a plane, as when a tennis ball bounces on a racket. We show that two regimes can be observed, depending on the strength of the force. At low forces, the ball makes contact on a flat disc. When this force gets bigger, the ball inverts itself on a cap, and the contact is limited now to the circular ridge in between the inverted and the non inverted part. The details of the geometry of the ridge are deduced from an analysis of the elasticity equations. Finally, we discuss, again by using the same elasticity equations, the problem of buckling of a 1

2

B. Audoly & Y. Pomeau

delaminated film. As it occurs quite often, a film coating some bulk material is under compression because of the way it has been deposited. This film may relax the compression by buckling out of the surface of the bulk material. In many instances a very specific pattern for the buckled film is observed, the so-called telephon cord delamination. We show that this may be explained as a result of a secondary bifurcation of a tunnel like structure of delaminated film, the Euler column. 2

Differential geometry of 2D manifolds

The equations for the elasticity of thin plates (FvK equations later) were de rived at the beginning of the twentieth century by Foppl and they are notorious for their complex nonlinear structure. Only recently various investigations put in evidence the possibility of getting explicit solutions in various limits that may be put globally under the heading of large deformations. Actually, those solutions rely heavily on the connection between the FvK equations and the underlying geometry. One central question in this geometry of surfaces, closely linked to elasticity problems, is to find the conditions for a given surface to be isometrically deformable. By this, we mean a deformation leaving unchanged the (intrinsic) distances measured along the surface. If one thinks of a piece of paper this intrinsic distance is just the length of a line drawn between two points on the paper. This length remains the same when the paper is rolled in one way or another, but without tearing, whence the name "intrinsic". Al though the definition of this intrinsic length is relatively straightforward in the present case, it becomes rapidly far more subtle when higher dimensions spaces are considered, and even for non planar 2D surfaces (like the surface of a sphere for instance). Riemannian geometry is the geometry of surfaces (and their generalization to higher dimensions, the so-called manifolds) such that the distances are invariant, independent on the coordinates chosen on the surface itself. That this is a crucial question in elasticity theory is evi dent when noticing that elastic energy precisely accounts for the amount of stretching occured by the material under the deformation. This stretching is measured by how much the distances between material points vary. In the present section, we consider the geometrical problem only, and we shall deal in a rather casual way with deep results of differential geometry related to this question of deformation of surfaces. Far more elaborate presentations of this topic (necessary anyway when dealing with manifolds of dimension higher than 2) can be found in [1]. The problem we shall look at is the following one: under what conditions is it possible to find a one-to-one map between a plane and a surface given

Elasticity and Geometry

3

by a Cartesian equation z = Z(x,y), without changing the lengths along the surface? Later, we shall also examine the existence of an isometric, one-toone mapping between two given surfaces. We are looking for a local map of a point of coordinate (x, y) in the horizontal plane to the point of coordinates x' = x + u(x,y), y' = y + v(x,y), z' = Z(x,y), that is situated on the surface. The functions u(x,y) and v(x,y) define "practically" the mapping under consideration. The constraint (imposed on u and v) is that the length element along the surface is the same as the length element on the plane, that is that ds'2 = dx'2 + dy'2 + dz'2 = ds2 = dx2+

dy2.

The orientation of the tangent plane of the surface at the origin (x, y) = (0,0) can be chosen arbitrarily with the help of a rigid-body rotation. We will therefore assume that it is horizontal. Then, the mapping is close to the identity near the origin. The Taylor expansion of Z(x, y) is quadratic in x and y near the origin, and we expect u and v to be small (actually, u and v are generically cubic mx,y near x = y = 0). Expanding dx12 and dy12 at first order in u and v, and at second order in Z (see explanation below), one gets: ds12 = d(x + u(x, y))2 + d(y + v(x, y))2 + dZ(x, y)2

, .du

nJ

+2dxM

d-y

dv +

rx +

dZ8Z^

^^-

Now the condition of invariance of the length element under the mapping becomes the condition that ds12 is the same quadratic form as ds2, which yields three conditions, one for the coefficient of dx2 to be one, another for the coefficient of dy2 to be one too, and the last one for the coefficient of the cross term dxdy to vanish: du l,dZ,2 n

d-x + 2^-°'

&

^ l,dZ,2 n ^+2(¥)2=0,

(3)

du dv dzaz md

n

dy- + 8-x + dx-dy- = °-

&

As we aim at eliminating u and v, there is one more condition than the number of unknown functions (three versus two), and one condition has to be satisfied for the existence of solutions of (2,3,4). This is to be imposed to the function Z(x,y), a data in the problem. It is obtained by deriving the first equation

4 B. Audoly & Y. Pomeau

twice with respect to y, the second one twice with respect to x and the last one once with respect to x and once with respect to y. Subtracting now the last result from the sum of the first two, the u's and u's cancel out and there remains an equation for Z only: 6PZ_&Z__ ( 82Z dx2 dy2 \dxdy

= 0.

(5)

This has a simple geometrical interpretation. Let us write Z(x,y) in the coordinate system diagonalizing its Taylor expansion near x = y = 0:

where ii lt 2 are the so-called principal radii of curvature of the surface at x - y = 0. Therefore, the equation (5) amounts to JJ^J- = 0, or equivalently to that at least one of the radius of curvature is infinite. Surfaces such that this holds true everywhere are called developable. When this condition is verified, integration of (2,3) u{x,y) « - ^ and v{x,y) ss - ^ . It is a straightforward exercise now to get by the same method the exis tence condition of an isometry for two smooth surfaces of Cartesian equations z = Za(x,y) and z = Zt,(x,y). One can take those two surfaces as tangent to the horizontal plane at x = y = 0, then redo the same calculation as be fore, but by imposing that the length on the two surfaces remain the same under two mappings. Those mapping depend on two functions ua,b{x,y) and wa,ft(a;»y), and are mappings from the plane to surfaces a and 6, such that x'a = x + ua(x,y), y'a = y + va(x,y) and z'a = Za(x'a,y'a) and a similar set with the subscript b instead of a. Now one imposes that the two length el ements (dx'a)2 + (dy'a)2 + (dZ'a)2 and (dx'b)2 + (dy'b)2 + (dZ'b)2, are the same quadratic form in dx and dy, which yields: *!• + 1/dZ°\2 = *£* . 1(9Zt,.2 dx + 2[ 8x ' dx V dx' ' *!•+ +{ h^H.* =*! + 1(^2 dy 2 dy) dy*2(dy), and

dUa

dy

dVa

dx

dZa

dZa

= dub dx dy dy

9vb

dx

dZb

d

h dx dy '

Because of the obvious similarity of these equations with the one of the pre vious case, one may use the same method to get rid of the functions ua,j, and ua,fc. One gets at the end that the Gaussian curvatures of the surfaces a and


5

b have to be the same:

iPZacPZa _ (d2za\2 dx2 dy2

\dxdy)

fd2zb\2

d^d^_ dx2 dy2

\dxdy)

'

W

This is the so-called Theorema egregium of Gauss (meaning approximately "outstanding", or "out of the crowd" theorem). This theorem is sometimes said as showing that the Gaussian curvature is a bending invariant: suppose that one can deform the surface isometrically ("bend it"), then its Gaussian curvature must remain the same at every point. This property is obviously a constraint on isometric deformations, but it is still in general a difficult ques tion to know if non trivial isometries exist for a given surface. For instance, a plane or a cylinder are deformable surfaces, but not a sphere, nor even a convex surface (when the edge is attached). In the coming two subsections, we shall expose two questions of differential geometry, the first one having to do with some properties of the developable surfaces, something that will be useful later on for the elasticity of thin plates, the next one will have a more mathematical bent and aims at showing an example of application of the ideas of differential geometry in a well defined case, the so-called Poincar6 half-plane. 2.1

Developable surfaces

By definition, such a surface may be mapped on a plane without stretching, and it is C2 smooth (an important assumption). Let us state first the Theo rema egregium in its general form (actually we stated it in the case of almost horizontal surfaces). Its extension is almost trivial, because it only requires to write the Gaussian curvature in an arbitrary system of coordinates. This can be done in a number of ways, and the final result is that the product of the principal curvature (or inverse of the principal radius of curvature) is equal to ri

\ —

{X V)

° ' -RlR2

— '5**"5y T

\~5xUy'

~ i+ (£)* + (£).•

/»•> (7)

which reduces to the left hand side of (5) when the tangent plane is horizontal. Therefore the algebraic condition that the Gaussian curvature vanishes is always

dx2 dy2

\dxdy)

~

6 B. Audoly & Y. PomeaM

The algebra will be made simpler later on with the notation

_ d*za d2zh V"»**\

dx2

dy2

d2zb d*za „ # % d2zb +

dx2

Qy2

'dxdydxdy'

so that G(x,v) =

I**L 2(l + (ff)! + (f)')'

A classical problem, called the Monge-Ampere equation amounts to find the unknown function(s) ° Z(x, y) such that G(x, y) is prescribed in a bounded domain for instance. In the case of zero curvature, the Monge-Ampere equa tion G(x, y) = 0 has an interesting general solution. Consider a one parameter family of planes of Cartesian equation: C(x, y\s) = a{s) + b(s)x + c(s)y,

where a(s), 6(a) and c(s) are smooth arbitrary functions of a parameter a. The envelop of this family of planes is the surface tangent everywhere to one plane in the family. As a first result, we show that this surface is tangent to those planes along straight lines (the generatrices). Consider two planes with neighboring indices, a and a + 6s, 8s small. Those two planes cross along a straight line, intersection of planes of equation z = a(s) + b(s)x + c(s)y, and

z = a(s + Ss) + b(s + Ss)x + c(s + 6s)y.

Take the difference between those two equations and divide by 6s, then one obtains that, in the limit of a vanishing 6s, the limit line of intersection of the two planes has Cartesian equation: z = a{s) + b(s)x + c(s)y, da db dc

^

dl + dlX+d-sy = 0-

(8)

W

These are two Cartesian equations of planes, showing that the envelope of the family of planes must include generically the straight lines whose equation is obtained in this way. It remains to show that the surface so generated has zero Gaussian curvature. This is shown directly, by computing G(x,y) as given by (7). The calculation is not straightforward, and so we shall decompose "There might be more than one solution: think of the case of zero Gaussian curvature.


7

the main steps. Notice first that equation (8) can be considered as giving s implicitely as a function of x, y. By differentiation with respect to x, one gets: fk = Ms) + — d(Q(3) + b(s)x + c(s)y) ( 10 ) dx dx ds and a similar equation for the derivative of z with respect to y. Prom the second equation (9), M*'+ \^)x+c\siVI — Q whenever one takes points on the surface tangent to the family of planes. Accordingly | | = b(s) and |^ = c(a). Therefore, d2Z dbds dx2 ~ dsdx' d2z dcds dy* ~ds"dy' d2z dbds dcds a a a a dydx ~" dsdx ~ ~ dsdy Putting this into the definition of [z,z]:

^'^-'[dxidy2

\dxdy) ) '

one gets that [z, z] is zero for a surface that is the envelop of a one parameter family of planes. This shows that the general solution of the Monge-Ampere equation G(x, y) — 0 = [z(x, y),z(x, y)] (that is for a zero Gaussian curvature) is z = £(x,y\s(x,y)), where s(x, y) is given by (9). Let us notice at this point that the surface found in this way in not necessarily physically representable by a folded sheet of paper, as it may cross itself, along either a line or at a conical point wherefrom a bundle of generatrices stems. 2.2 Geometry of the Poincari half-plane This subsection is to show on a concrete example how differential geometry works. This example is the one of the Poincari half-plane, that is the upper half plane, x arbitrary and y positive. This half plane is endowed with the metric

ds2 = * ^ ± * 1 . 2

(12)

y This metric is invariant under dilations centered on the x axis and is actually related to a projective metric in higher space dimensions [2]. The mathematics of the Poincari half-plane is an immense field of investigations and we shall


content ourselves with a few simple remarks aiming to show how differential geometry works in a specific case. The first simple remark is that this metric cannot be realized by a surface sitting above the half plane, since for large j/'s, the length element becomes too small to be written as ds2 = dx2 + dy2 + dZ2, where Z(x,y) is some function of x and y. A fundamental notion in differential geometry is the one of geodesic line. By definition, a geodesic has the shortest arc length between two points. This holds true for a positive definite metric, as the one under consideration. For a non positive definite metric, one replaces this condition of shortest distance by the condition that this distance is extremal in the variational sense. The geodesic lines of the Poincare half-plane are called horocycles, and they minimize the arclength J * "*" v . The minimization is easier by assuming that the Cartesian equation of the horocycle is given by x(y), and one gets as Euler-Lagrange condition, the so called Jacobi equation:

5 (TfTTPs) = °'

(13

>

where x' = ¥■. This can be solved at once to give the general parametric equation for the horocycles: x — -Rcos(j> + x0 and y = R sin <j>, where R is an arbitrary positive constant and an angle in [0,7r]; XO is the center of the semicircle drawn by the horocycle. There are two other remarkable properties of the Poincare" half-plane. The first one is that it has constant negative Gaussian curvature. In the present framework, one cannot define Gaussian curvature as we did before, because there is no explicit realization of the Poincare half-plane as a real 2D surface. Actually the Gaussian curvature may be defined intrinsically as well, through the general formula giving the total rotation of the tangent to a closed curve (the Gauss-Bonnet theorem). By definition, there is no rotation of the tangent along a geodesic. Therefore, it is easier to consider this total rotation along a closed curve made of geodesic arcs. In ordinary plane geometry, the total rotation of the tangent to a smooth closed curve is 2ir. For a polygon with straight edges, this is still the same 2ir if one blunts a bit the vertices in such a way that the tangent along its vertex (of index j) rotates by at - it where <Xj is the outer angle of the vertex. Therefore Ylj(aj - 7r) = 27r for any closed polygon, a familiar result indeed. Take a triangle, this yields £ \ aj = 5n or X)j(27r - aj) = IT, showing that the sum of the inner angles 27r - a, is n.

Elasticity and Geometry 9

Figure 1. Geodesic polygon in the Poincarf half-plane.

We borrow the proof of the Gauss-Bonnet theorem for the Poincare half plane from C.L.Siegel [3]. Consider a polygon with its side made of arcs of horocycles, and compute the area fi inside this polygon. The area element consistent with the metric is ^p-. The area inside a polygon is just the integral il = / ^ ^ extended to the inside of the polygon. Integrating over y turns the area into a sum of line integrals along each geodesic arc bounding the polygon:

»-?£-(*)• where C* is the arc of index k bounding the polygon. The line integrals over Ck are computed by using as variable the angle 0 that was introduced when solving Jacobi's equation for the geodesies: dx = Rsin<j>d and y = Rsm<j> and the line integral becomes H = £ A f£ -d = - £ ^ ( 7 * - 0k), where 0k and 7* are the values of at the ends of the arc C* (see fig. 1). Now one notices that (7*; - 0k) is precisely the amount by which the normal has rotated from one end to this arc to the other. The sum ]Cfc(7* - 0k) is 2;r minus the sum of the angles necessary to turn the normal at each vertex when going from one arc to the next. These angles read also (a* - ir)'s, with a* outer angle at vertex k: - f t = £ A ( 7 * - 0k) = 2n - £ * ( Q * - n). This is a particular case of a more general formula, the Gauss-Bonnet formula that relates in general the integral of the geodesic curvature along a closed curve to the integral of the Gaussian curvature inside the enclosed area. This formula reads, for a polygon bounded by geodesic arcs: V(ajb - TT) + / KGdu) = 2TT, k

(14)

Jn

where KQ is the Gaussian curvature, du> the area element. This formula can be written in a more general form for a general (non piecewise geodesic) closed

10

B. Axtdoly & Y. Pomeau

curve. The Gauss-Bonnet formula in its form (14) shows that the Gaussian curvature of the Poincare half plane is (-1). Another important property of the Poincare half plane is that it has a continuous group of isometries, the so-called modular group. Its general ele ment depends on three real variables (it is actually easier to use four variables with one constraint). The general transformation reads: z->z'

,

az +b = ——cz + d

, , 15

In (15), a,b,c and d are real numbers such that ad - be = 1. The latter condition puts the map into a kind of canonical form. Actually, the map is not changed by multiplication of all four parameters a, 6, c and d by a non zero real constant, a multiplication that changes the value of ad - be. Setting this to 1 selects one representative among all possible choices of (a, 6, c, d) for the same transformation. Because the map z -> z' is analytic it conserves the angles (it is a "conformal" map). That it is a group is checked directly: the products of two maps with parameters a, b, c, d and a', b', d, d1 has parameters a" = aa' + be1,6" = ab' + bd',c" = ca' + dd,d" = cb' + M, and one has a"d" - b"c" = (ad - bc)(a'd' - b'c'), which shows that the condition for the unit determinant is conserved by the map. The unit element is a = c = 1, b = d = 0. The inverse is computed by getting z as a function of z'. The metric (15) comes naturally into play in the Poincare' half-plane. As we show, it indeed leaves invariant the length element defined in (12). It is simpler to show it for the transformations close to the identity, that is for maps of the form z' = z + ef(z), with e small real and f(z) function of z to be defined. First of all, one notices that because of its form, the metric is obviously invariant under translations (small or large) in the x direction as well as under dilations from the origin. The corresponding map, when small would correspond to f(z) = 1, for translations, and to / ( z ) = z for dilations. The last possibility for f(z) corresponds to a = d = 1, b = 0, and c = - e , namely to /(z) = z 2 . The Euclidean length element becomes: dx'2 + dy'2 = dz'dz1 = dzdz(l - e(z + z)), although the denominator y2 becomes: ^2=~4(Z'-Z')2 * " 4 ( z - z ) 2 ( 1

+

€ (

*

+ I ) )

'

which shows that, to first order in e, dx \dV i s left unchanged. We shall end this short account of some elementary properties of the PoincarS half plane by mentioning that it underlies the theory of spinors in


quantum mechanics [4]. Also, we notice that the associated metric has a non trivial isometry group. The general question of finding the full isometry group associated to a quadratic form seems to be quite difficult * (besides the now obvious condition that the map has to be between points with the same Gauss curvature). Think for instance of the half upper plane with the indefinite metric ds2 = dx ~*y . The geodesies become pieces of hyperbolae now, but it is not obvious that the isometry group is bigger than its obvious part, the dilations and the translations in the x direction. However, there is chance that the isometry group is big, since Gauss curvature for this indefinite metric (meaning that the square length can be positive, zero or negative) has constant positive Gauss curvature equal to + 1 . 3

Thin plate elasticity

The Theorema egregium gives the condition under which a surface is mapped into another surface by keeping the distances invariant. This has obvious application in elasticity of thin plates and shells, since one expects naturally that these deformations keep at its minimum the energy of strain: if the length along the ideal zero thickness surface is unchanged, an eventual energy of deformation will be due to the small thickness of the plate, that is to its 3D structure. This elastic energy will be likely some sort of small correction in a systematic expansion in the small parameter represented by the plate thickness (or, more precisely by the aspect ratio). However, the mechanical problem is not such a straight application of the geometrical concepts, for two reasons. First the energy of deformation may involve at the same order inplane stretching effects, and (3D) bending effects (which are absent in a purely geometrical description). The Gauss-inspired differential geometry relies upon the crucial assumption of smoothness of the surface to establish the condition of isometric mapping (actually C 2 -smoothness in technical terms, meaning existence and continuity of the tangent plane). However it happens quite often that a sheet or a shell under strong deformation will actually minimize its elastic energy by becoming singular in the sense of the 2D geometry of surfaces, the regularization being due to the three dimensional structure: for instance, a crumpled plate will get a radius of curvature of order of magnitude of its thickness h, something that would become a singular ridge in the zero thickness—ideal plate limit. If one crumples a piece of paper (or better a 6

An interesting, if not important, question related to this topic is the formation of wrinkles on human skin. Facial wrinkles may be seen as due to the fact that no isometric sliding of the skin exists on the surface of the skull.


sheet of transparency, a much better material than paper for this purpose) one can notice crescent-like marks (or scares) left on the paper, once it is uncrumpled. Those scares show up at very localized places, manifesting a focusing of stresses such that the elastic limit of the material (polymetacrylat for a piece of transparency) has been so overpassed that some irreversible phenomenon took place there, leaving a scare on the otherwise smooth sheet. A very simple observation explained in [6] shows (although in a rather non trivial way) how differential geometry and elasticity are related there. This kind of remark has more than an anecdotical character, as it is also relevant for crash of cars, with the body is strongly deformed at well definite locations, due to the same phenomenon of stress-focusing. The folds on fabrics flattened by ironing fall in the same class. A plate is a piece of thin elastic material of constant thickness (this later condition could be relaxed at the price of great notational burden). In its rest state a plate is a plane. Although the equations of elasticity for thin rods were derived by Euler in the eighteen century, it took another century and a half to derive the equation of elasticity of thin plates, something done in 1905 by Foppl. Since those equations are widely known as the von Karman equations, we shall call them afterwards the FvK equations. They allow to compute consistently the energy of deformation of a thin plate of elastic material in the Hookean case c . By definition, a thin plate is a piece of 3D elastic material of thickness much less than its width in any other dimension. In this limit of a large aspect ratio, and by assuming that the plate remains almost horizontal, the first FvK equation for an equilibrium deformation read : h3E j^--Â2C-MC,X]

= 0,

(16)

where x is the so-called Airy potential and £(x, y) the (small) deviation of the plate out of the z = 0 plane (x, y are the Cartesian coordinates in this plane), h is the thickness of the plate and E {/a ) its Young modulus(/Poisson ratio). Moreover A 2 is for the bilaplacian (that is the square of the Laplacian):

dx* c

dy*

dx2dy2 '

The Hookean approximation forbids one to consider the irreversible deformations alluded before. It would require to take into account irreversible phenomena in elasticity, something that is much beyond the scope of this work. Note however that this occurrence of scares, for instance, is a way of study, at least indirectly, the irreversible behavior of solids without breaking large pieces of material.


13

The equation (16) is to be completed by the second FvK equation: A 2 x + £[C,C1 = 0.

(17)

The symbol [U, V] in (16,17) with U, V either ((x,y) or xix,y) before: 1

'

J _

dx2 dy2

+

dx2 dy2

dxdydxdy'

is defined as

(

'

The FvK equations (16,17) describe the elasticity of a plate. We shall examine now three basic questions related to the FvK equations, the question of the boundary conditions being left to the next section: i) These equations are the Euler-Lagrange equations for an energy func tional (being derived through a consistent schema from the general equa tion of elasticity, the Euler-Lagrange structure carries over). ii) Although this looks a simple question, the orders of magnitude consis tent with the FvK equation are not so trivial. The main point we are going to show is that, for large deformations, the flexural term (that is i2(i-^-s)A2C in (16)) is negligible, because it involves the largest power of the small thickness h. Without this term, and for convenient boundary conditions, the energy minimum is a purely geometrical problem that reduces itself to the one of developable surface. But this last one may have no smooth solution, so that a more complicated "basic" (= without bending energy) solution has to be found, where the flexural term may become locally relevant (the radius of curvature of the surface becomes locally much smaller than its average value, a typical boundary layer situation). iii) There is a rather tight connection between the proof we gave of the Theorema egregium and the structure of the FvK equations. This is shown below by deriving that part of the FvK equation that takes into account the stretching energy only. Moreover, the derivation presented below will permit to introduce in the next Section the boundary conditions for the FvK equation in a kind of rational way. 3.1

Euler-Lagrange functional

As it can be verified after integrations by part, and by skipping for the mo ment the boundary terms (considered in next Section), the FvK equations are


deduced by Euler-Lagrange variation with respect to \ and C of the functional:

Z

h?E

1

(19) i d : r d y [ 2 4 ( l - ^ ) ( A C ) 2 ~ 2E ( ( A * ) 2 " X [ C ' C 1 ) ] This integral is over the whole area of the plate A, again assumed to be close to the plane 2 = 0. One can get rid of the Young modulus in (17) by redefining a scaled Airy potential x' — % so that the functional to be studied becomes:

F'(C,x')=Eh f dxdy JA

h

24(1 - a2)

(AC)2-iU(Ax')2-x'[C,C])

(20)

As this functional does include a cubic term, x'[C C]i w *th the largest power (any other term is formally quadratic in either £ or x'), this energy has no well defined sign and cannot be bounded from below (or from above). A simple trick circumvents this difficulty: let us consider x' as a function of [£, Q (that is of the Gaussian curvature as it enters in the second FvK equation (17)). Then, once this is done, the Airy stress function has disappeared from the energy integral, and this one depends now only on the shape of the plate given by {(x,y). FVom the algebraic point of view (that is by forgetting that £ and x' are functions of x and y, taking them as ordinary real variables), one can see the formal elimination of x' as what would be done when computing the (local) minimum of a function F"(C, x') = 2 | — V + *'C2> w i t n *'» C ordinary real variables, and a a coefficient (its role will be explained shortly). Indeed F" has no absolute minimum because its highest power is the cubic x'C2 that can be made arbitrarily large negative or positive, and much larger in absolute value than any other term in F" by taking x' large negative and £ large, positive or negative. However, this function has a local minimum, found by solving for \' the equation î- - 0, which gives x' - C2> which, once put into F" gives F" = ^j- + |C 4 i that has one or three local minima, as a function of C, depending on the sign of a: if it is positive, then the unique minimum of F" is at ( = 0, while if a is negative, the extremum of F" for < = 0 becomes a maximum, and two new minima at C2 = Vxx, uyy and uxy in function of x- The result is: Ox* = - ^ = 2(A + 2fi)uxx + 2Au;„v, dy* a

w = -Q^ = 2(A + 2»)uyy + 2Aw«,

(30) (31)

2

axy

=

&X ~dx^

= 4fMJxv

(32)

-

One deduces from this system the components of the strain u in function of the second derivatives of x,,

_ (A + 2/i)x, v l ; -Ax ,XX

"xx ..

/nn\

'

(33)

'

(34)

_ (A + 2/i)x,»» - \x,w

"- ~

»°d

MAT^) SMAT^)

«„ = -*g*.

m

,

(35)


Indeed those three equations begin to look like the one we met before when proving the Theorema egregium: this theorem states the condition under which the components of the strain are all zero, which is equivalent to say that the strain w vanishes, that is precisely a condition deduced from the cancellation of the three components of u just written in (33, 34 and 35). When proving the Theorema Egregium, we have used the identity o2uxx d2u}yy d2uxy _ d2z .2 „d2zd2z K dy2 dx2 dxdy dxdy' dx2 dy 2 Therefore, by applying the same combination of second derivatives to the right hand side of the expression of u in terms of the derivatives of x, one obtains an equation relating the Airy potential (actually a linear combination of its fourth derivative with respect to x and y) and Gaussian curvature. One gets the unpolished expression: d2Z% dy2

[

d2Z2y dx2

2

2d

(Z<xZ,v) dxdy

1

=

8/I(A + /J)

ix+2 +

^ w]

_ 2 A _2^ _ +4(A v 8x dy2

+

(36)

^ ) , a 24 x 2 ^'dx dy _

which becomes, after proper rewriting: A2x

+ E[Z,Z] = 0,

with E = 8 f f i ^ ■ This is the second FvK equation (17) The non bending part of the first FvK equation is derived along very similar lines by writing the condition of stationarity of Uei under variations of Z. This gives as a first result: — (2AZ x(uxx

+ u)yy) + ifxZy(u)xx + uyy) + 4/iZ, z w xs ) = 0.

(37) Now the first FvK equation follows by substituting into (37) the expression of the components of u> in terms of the second derivatives of x- The final result is bilinear in Z and x and is just the part of the first FvK equation not involving the bending, that is [x, Z] = 0. 3.4

Thin shells: an example

We discuss in this Section the contact of a thin elastic shell (actually either a ping pong or a tennis ball) with a rigid plane. Experiments show a first or der/discontinuous transition between two different configurations: the first is characterized by a flat contact between the shell and the rigid plane, whereas

24


in the second case the shell inverses itself and makes contact with the plane along a circular ridge only. This striking phenomenon can be explained, al though in qualitative terms, by comparing the energy of deformation of the two configurations. Below we shall explain first this phenomenon in qualita tive terms only, and then we shall sketch a more quantitative approach, that rather surprisingly uses the FvK equations. This is a bit surprising because the FvK equations concern flat plates without any rest curvature, although a ball is what is called a shell in elasticity theory, because of its equilibrium curvature. In their book on Elasticity Landau and Lifshitz present the solution of the problem of a spherical shell strained by a normal force pointing inwards, a solution due to Pogorelov [8]. When the intensity of the force is small, the deformation is localized near the point of application and grows linearly with the force. For large forces a circular fold appears around the point of application. In the latter case, the scaling laws for the radius of the fold can be deduced from order of magnitude estimates of the various contributions to the elastic energy. As shown by Pogorelov, if the applied force is F, then the deformation, e is proportional to the square of F for F large: F

Eh2 — e for e < h,

(38)

R/,5/2

F

5—e 1/2

for c > h,

(39)

where E is Young modulus, R the radius of the shell at rest. In the case studied by Pogorelov, the transition from linear to quadratic behavior of F at increasing e is continuous, and happens roughly when c is of order h. This concerns a force applied with a sharp point of radius of curvature much less than the shell radius. On the contrary, we consider below the opposite case, namely when the force has a wide support, for instance when a ball is crushed between two parallel plane, or the dynamics of a tennis ball bouncing on a racket, both situations similar to the classical Hertz contact problem. The experiments, as reported in [5] show two regimes, depending on the strength of the applied force: i) For weak forces, the deformation is small as well, and part of the ball gets into contact with the rigid plate along a disc. ii) For larger forces, the disc of contact unfolds and reverse itself to become an inverse spherical cap, the contact with the rigid plane being now along a circular ridge, that separates the inverted cap from the rest of the


Figure 3. Geometry of the ball at large forces in configuration II,left, and close-up view of the circular fold, right.

sphere (see fig. 3). The elastic energy is stored in the latter case almost exclusively in the fold. Later on we shall call the two configurations (I) and (II). The transition takes place for a deformation (as measured by the change of distance to the center of the sphere in the radial direction) of order 2/i. The dependance of the force on the deformation is like F ~ e, for e < 2/i and F ~ e 1 / 2 for e > 2/i; which shows two different behaviors. Furthermore, an hysteresis is observed as the force is first increased and then decreased, which implies a discontinuous transition from one regime to the other, as observed too. A detailed interpretation of the experiments would require to take into account various irreversible phenomena in the material the balls are made of. We shall not consider this question here, and limit ourselves to the discussion of the purely elastic and reversible phenomena. To estimate the elastic energies associated with configuration (I), one no tices that this energy is made of two pieces: first the flattening of the spherical cap changes the lengths measured along a piece of the shell, which goes from spherical (the rest state) to a flat disc. This involve a stretching energy, be cause the Gaussian curvatures everywhere changes from R~2 to 0. Secondly, a fold is created along the edge of the contact region, the contribution of which has to be considered. The stretching contribution to the energy is computed as usual, as the integral of the square of the strain over the volume. The radius of the disc of contact is r = y/2Rt for e small. The straining of the length is estimated as fol lows. The unperturbed cap, bounded by a circular ridge of radius r is a spher-

26


ical disc of radius Ra along the surface of the sphere, with sin a = ^ . This curvilinear distance Ra becomes R sin a once the cap is flattened to become a disc of radius r. Therefore, the strain—the derivative of a displacement—once integrated along a radius of this disc, is equal to the difference between the unperturbed and the perturbed length, namely to /?((sina) — a) ss -R9*-The gradient is approximately this displacement divided by r, which gives JJH ^ Bsl „ will be decomposed into a contribution due to the initial compression ) belonging to some Hilbert space, which de scribes all the physical properties of the system. Its evolution is given by the Schrodinger equation:

«4S2>.H(OW(q) is not directly observable in quantum mechanics. In general - ac cording to the standard Copenhagen interpretation of quantum mechanics the result of a measure is some diagonal element of an Hermitean operator, something like (ip\0\ip) [3]. The physical processes involved in an experimen tal measurement are quite subtle, difficult and interesting, but beyond the

Quantum Chaos 39

subject of these lectures. It is also the subject of a vast litterature [4]. I will not consider this problem and restrict to a purely Hamiltonian evolution. The time-evolution operator U(t',t) is by definition the linear operator mapping the state \rp(t)) onto the state \ip(t')). It obeys the following equation (which is equivalent to Schrodinger equation): ih9-^-=H{t')U{t',t).

(5)

U(t',t) is the major object for studying the quantum dynamics. Because H is an Hermitean operator, U(t',t) is a linear unitary operator. An imme diate consequence is that the overlap between two states is preserved during the time evolution. Indeed, one has:

to(Ohfe(f)> = (Mt)\uHt',t)u(t',t)\rp2(t)) = MWItfeW)

(6)

which implies that two "neighbouring" states will remain neighbors forever. Because of linearity and unitarity, quantum mechanics cannot display any sensitivity on initial conditions, hence cannot be chaotic in the ordinary sense! However, the previous statement must be considered with care. Indeed, classical mechanics can also be seen as a linear theory if one considers the evolution of a classical phase space density p(q, p, t) given by the Liouville equation [2]:

where {,} denotes the Poisson bracket. The fact that we obtain both in classical and quantum mechanics a linear equation of evolution in some space just implies that the above argument on linearity in quantum mechanics is irrelevant. Discussions on subjects like "Is there any quantum chaos?" are in my opinion completely uninteresting because they focus on the formal aspects of the mathematical apparatus used. We will here define quantum chaos as the study of quantum systems whose classical dynamics is chaotic. The questions we would like to answer are thus: • What are the appropriate observables to detect the regular or chaotic classical behavior of the system? • More precisely, how the chaotic or regular behaviour expresses in the energy levels and eigenstates of the quantum system? • What kind of semiclassical approximations can be used?

40 D. Delande

These are the questions discussed in these lectures. I will only present selected topics, forgetting lots of interesting questions and relevant references. Thes questions of course go towards an intrinsic definition of quantum chaos not refering to the classical dynamics [5]. Thus, the problem of quan tum chaos is essentially related to the correspondance between classical and quantum dynamics, the subject of semiclassical physics. 1.3

Semiclassical dynamics

The whole idea of a semiclassical analysis is to obtain approximate solutions of the quantum equation of motion (the Schrodinger equation) using only classi cal ingredients (trajectories...) and the Planck's constant ft. For a macroscopic object, our common knowledge is that an approximate semiclassical solution should be very accurate. Technically, this is true because ft is much smaller than any classical quantity of interest (such as the classical action of the par ticle). One often refers to the "correspondance principle" as an explanation. However, this is a very vague concept which is usually not clearly stated, not proved and whose conditions of validity are not discussed. Actually, it is so vague and qualitative that it should be rejected. Part of these lectures are devoted to a serious scientific discussion of this issue, using the modern knowledge on classical chaos. In order to make the connection between classical and quantum quantities, it is useful to define the Wigner representation defined as [6]:

W

^

P)

= ( 2 ^ / * (\2) exP ( - ^ S r H )

\

H^H, - c o , all real eigenvalues disappears (except 0 in the second problem), leaving place to the entire real axis forming the essential spectrum : for a real ^ 0 the operator ( PQ, or at a distance of order 1 from the imaginary axis, then the estimate (7) allows us to find a center manifold (see [15], [19], [22]). Roughly speaking, all "small" bounded continuous solutions taking val ues in HJ, of the system (1) for values of the (multi) parameter p near po, lie on an invariant manifold M^ which is smooth (however we loose the C°° regular ity) and which exists in a neighborhood of 0 independent of p. The dimension of M,! is equal to the sum of dimensions of invariants subspaces belonging to pure imaginary eigenvalues for the critical value po of the (multi)parameter.

The Water- Wave Problem as a Spatial Dynamical System

83

In other words, the modes corresponding to eigenvalues far from the imagi nary axis are functions ("slaves") of the modes belonging to eigenvalues near or on the imaginary axis. In addition, the reversibility property leads to a manifold which is invariant under the reversibility symmetry S. The trace of system (1) on M^ is also reversible under the restriction So of the symmetry S. At this point we should emphasize that the physical relevance of this reduction process is linked with the distance of the rest of eigenvalues to the imaginary axis. So, this validity is going to 0 when the thickness of the bottom layer increases, and in such a case we have to think to another technique (see section 6)! 4

Basic codimension one reversible normal forms

Once the problem is reduced to a reversible ODE, we need to examine various possible critical situations. This basically depends on how many eigenvalues lie on the imaginary axis, at criticallity \i = fio- The reversibility property reduces the cases of interest to the following: (i) LUo has only a double 0 eigenvalue on the imaginary axis, (ii) Lpa has only a double 0 and a pair of simple pure imaginary eigen values on the imaginary axis, (iii) LM0 has only a pair of double pure imaginary eigenvalues on the imaginary axis, (iv) LM0 has only two pairs of simple pure imaginary, strongly resoning, eigenvalues on the imaginary axis. There are other problems of interest when more than 4 eigenvalues may lie on the imaginary axis, as it is the case for problems with several layers, and with additional parameters like surface and interfacial tensions. There are also other interesting cases, for instance when 0 is a quadruple eigenvalue. This would correspond to a codimension two singularity (occurs in playing with two parameters), so it is not a generic case (see [7] for a study of this case). We shall not consider such cases here, whose study follows the same stream. In addition, for a problem such as the one of section 2.2, we always have a 0 eigenvalue which should be counted in addition to the above ones, with its multiplicity (2). For each of these cases this corresponds to a particular critical set in the parameter space, and we denote by v the bifurcation parameter, representing a (oriented) distance of y. from the critical set of (JLQ. The basic technique we use to study such cases is first normal form theory (see for instance chapter 1 of [6]) to simplify the reduced system up to a fixed order. This corresponds to a suitable choice of variables (after a polynomial

84 G. Iooss

B

A -3v/2a

Figure 3. Phase portrait of vector field (8) for u > 0, a < 0.

change), and this allows to recognize all types of bounded solutions: a re markable fact here is that in all cases (i) to (iv), we obtain integrable normal forms. This means that for the system truncated at any fixed order, we know all its small bounded solutions. We give below more details of these solutions.

4.1

Case (i)

Here the center manifold is two dimensional. Let us define by (A, B) the (real) coordinates (or "amplitudes"). Then we need to know how the reversiblity symmetry S 0 acts on (4, B). There are two theoretical possibilities: (A, B) -> {A, -B) or (-A, B). In all water-waves problems the first case holds, and then the normal form, truncated at quadratic order, reads

where one can compute explicitly the coefficient a. Here, for v > 0 the critical eigenvalues ±^/u are real, while they are pure imaginary for v < 0. The vector field (8) is integrable, and its phase portrait is given at figure 3, for v > 0, and a < 0. We observe that there is a second steady solution, corresponding to fiat free surface and interface ("conjugate flow"), there are periodic solutions with various amplitudes, and there is a homoclinic solution to 0, corresponding to a "solitary wave" of depression for problem (1,2). This problem was first solved in [1].

The Water-Wave

Problem as a Spatial Dynamical System

Figure 4. case (ii). Graphs of /K,H(.A)

4.2

85

for v > 0,o > 0.

Case (ii)

Here the center manifold is four dimensional. Let us denote by ±iq the pair of simple eigenvalues and define by (A, B) the (real) amplitudes and C the complex one, corresponding to the oscillating mode. Then we need to know how the reversiblity symmetry So acts on {A, B, C, C). Here again, there are two theoretical possibilities. In all our problems we have: (A,B,C,C) -* (A, —B, C, C) and the normal form, truncated at quadratic order, reads & = vA + aA2+c\C\2, %=iC(q + d1u + d2A),

(9)

where (real) coefficients a,c,dj can be explicitly computed. This system is indeed integrable, with the two first integrals K = \C\\ 2

H =B-

(10) (2/3)aA

3

2

- uA -

2cKA.

We show at figure 4 the various graphs of functions faî-A) = (2/3)aA3 + i/A2 + 2cKA + H depending on {K,H), for v > 0,a > 0. In this case, we have, in addition to the conjugate flow (as above), several types of periodic solutions, quasi-periodic solutions (interior of the triangular region in (K, H) plane, and homoclinic solutions, one homoclinic to 0, and all others homoclinic to one of the periodic solutions. We represent on figure 5, in the (A,B)

86 G. Iooss

Figure 5. case (ii). Bounded solutions for various values of H of (9) in the (A, B) plane, for v > 0,a > 0,0 < cK 0 the eigenvalues are at a distance ^/v from the imaginary axis, while, for v < 0, they sit on the imaginary axis. The vector

The Water-Wave Problem as a Spatial Dynamical System 87

Figure 6. case (iii). Graphs of fK H (u) in the (K,H) plane, for u > 0,92 < 0. Hm = v2/2q?,

K

2

H3/2

field (11) is integrable, with the two following first integrals: K = i/2(AB-AB), H = \B\2- I Jo

(13)

Q[v,u,K]du.

It is then possible to describe all small bounded solutions of (11). Indeed, we obtain

(*£)' =/«<W>. where fKM\A\2)

= 1q2\A\6 + 4(u + q3K)\A\* + 4H\A\* - AK2.

We show in figures 6 & 7 the various graphs of functions JK,H depending on (K,H), for v > 0,92 < 0, and for v < 0,g2 > 0, which are the most interesting cases. In such cases we obtain large families of periodic and quasi-periodic solutions and, for g2 < 0 a family of solutions homoclinic to 0 (solitary waves with exponentially damped oscillations at infinity, sometimes called "bright" solitary waves), while for q2 > 0 we have a family of solutions homoclinic to periodic solutions (as in case (ii)) which correspond to so-called "black" solitary waves (the amplitude is minimum at x = 0).

88 G. Iooss

Figure 7. case (iii). Graphs of // 0.

4-4

Case (iv)

This case corresponds to 2 pairs of simple eigenvalues {±iwj,j = 1,2) on the imaginary axis, where for v = 0 we have u>i/u;2 = p/q < 1, P and g being positive integers. In fact when v varies the eigenvalues move on the imaginary axis and this rational ratio is lost, but what is important is that our analysis stays valid for values of v near 0 such that the interval of values taken by the ratio between the two eigenvalues does not contain a rational number p'/q' with p' + q' < p-¥ q. Here the center manifold is four dimensional, and we denote by A\, A2 the twoômplex amplitudes. The reversibility symmetry So acts as: (Ai,A2) -*■ (Ai, A2). Strongly resonant cases are such that p = l,q = 2 or 3. "Reversible 1:2 resonance" denotes the case with u\lu)2 = 1/2, and we concentrate our analysis on this case. The normal form truncated at cubic order reads: dA\ — 2 +al2\A2\2) + ibiAiA2, (14) dx = iAi(u + aii/ + an\Ai\ dA2 = iA2(2u + a2v + a 2 i|>ii| 2 + a22\A2\2) + ib2A\, dx where all coefficients aj,6j,a i; , are real. This system (14) is integrable, with the following two first integrals: K = b2\A,\2 +

bl\A2\\

H = A\A2 +A\A2 -b^(a2 -2aiH4i|21 (26i)- (a 2 i - 2a 11 )|7l 1 | 4 + +(26 2 )- 1 (a 2 2 - 2a 1 2 )|.4 2 | 4 .

(15)


89

It is then possible to identify all periodic and quasi-periodic solutions, and homoclinics to periodic solutions, depending on the values of coefficients. 5

Typical results for finite depth problems

In previous section, we investigated the normal forms, and obtained various type of solutions that we might expect for our complete problems. For finite depth problems, thanks to the use of a center manifold reduction, we are now working on a reversible ordinary differental equation, and the above normal form is just the principal part of this ODE, whose higher order terms are not in such a nice form. The natural mathematical problem consists now in proving persistence results. In summary, the persistence of periodic solutions of the normal form can in general be performed, through an adaptation of the Lyapunov-Schmidt technique [9],[17]. The persistence of quasi-periodic solutions is much more delicate, and can only be performed in a subset of the 2 dimensional space of first integrals, where these solutions exist for the normal form (see a complete proof in [8], and another example in [9]). The persistence of solutions homoclinic to periodic solutions, provided that they are not too small, needs some technicity (runs well in the reduced 4 dimensional space), see for instance [9] for case (ii) and [10] for case (iii). About the method for computing all coefficients in such non trivial normal forms see [3]. Now, for the normal form of case (ii), there is a family of orbits homoclinic to a family of periodic solutions whose amplitude can be chosen arbitrarily small. Such a case (ii) has been investigated by many authors (see for instance [2], [20], [21] and [9] and references therein). There are homoclinic solutions to oscillations at infinity whose size is smaller than any power of the bifurcation parameter, corresponding to the fact that we cannot avoid such oscillations when we consider the full untruncated system. The extremely delicate aspect of exponentially small and still existing oscillations was proved by Sun [21] on the water wave problem (1,2), and for a wide class of problems is thoroughly studied by Lombardi in [17]. Moreover, despite of the fact that a solution homoclinic to 0 exists for the normal form (9), this is not true in general for the full system (see [18]), even though one can compute an asymptotic expansion up to any order of such an homoclinic (non existing) "solution"! For these results on homoclinics, when they exist, it should be mentioned that the decay at infinity is exponential. There are degenerate cases (codimension two situations) where this is no longer true. For example in the case (iii) when coefficient 92 is close to 0 (see [11]) it can exist in general (for v = 0) an homoclinic to 0, with a polynomial decay at infinity. It should be noticed that this phenomenon is in fact very different from the similar property of

90

G. Iooss

polynomial decay that we shall meet at next section. Both phenomena are due to very different causes.

6

Infinite depth case

We now consider the case where the bottom layer has infinite depth. This means in our model problems that for problem (1,2) we must replace b by 0, except in Xb which is the new parameter (i. In problem (1,3), we must replace e by oo and there are still two parameters \x = (X,p). Let us notice that, for most of physical situations, the infinite depth case is much more interesting than the finite depth one. Indeed, the domain of validity of our nonlinear (local) theory is limited by the distance from the imaginary axis of the rest of the spectrum (the non-critical eigenvalues) of the linear operator studied in section 3. It appears that this distance goes to 0 when the depth increases. Now, typically the length scale for water in problem (1,2) is of the order of one centimeter, so this means that for layers of more than few centimeters depth, many of the non critical eigenvalues are very close to the imaginary axis. It is then clear that our study becomes just academic! As generally, we need to study the worse limiting case, which is here the infinite depth case, and physical cases are in fact regular perturbations of this limiting case. We shall see below that this has dramatic consequences on the mathematical analysis!

6.1

Spectrum of the linearized problem

Passing to the infinite depth limit in the dispersion relations (5,6) we find much less eigenvalues. In fact all real eigenvalues disappear, except 0 for problem (1,3). However the real axis belongs entirely to the spectrum of L^ (see [12],[4],[13]). Indeed, for any non zero real a, the operator 61 — L^ is one to one but not onto (one says that the real axis forms the "essential spectrum" of LM). Its range is not closed in H, and the codimension of its closure is non zero (see [13]). It results that the spectrum of L^. crosses the imaginary axis at 0, and that we cannot use the center manifold reduction to an ODE. Despite of this awfull fact, we still have isolated eigenvalues of finite multiplicities outside of the real axis, and situations with pairs of eigenvalues on the imaginary axis as the ones described in cases (iii), (iv) of section 4. In addition, we still have the resolvent estimate (7), due to a good choice of space IHL


6.2

91

Normal forms in infinite dimensions

Since we cannot reduce our problems to finite dimensional ODE's, we still would like to believe that eigenvalues near the imaginary axis are ruling the bounded solutions. This is a justification for developing a theory of normal forms in separating the finite dimensional critical space, from the rest (the "hyperbolic" part of the spectrum, including 0). This leads to "partial normal forms", where there are coupling terms, specially in the infinite dimensional part of the system (see [4],[12]). However there are some additional difficulties: i) due to cases where 0 is an eigenvalue embedded in the essential spectrum, our technique uses the explicit form of the resolvent operator near the real axis, to detect the good continuous linear form which can be used for the projection on the eigenspace. ii) In space H the linear operator has not an "easy" (even formal) adjoint. This adjoint and some of its eigenvectors are usually necessary for expressing projections on the critical finite dim space. In our problems, we use again the explicit form of the resolvent operator near the (double) eigenvalues, to make explicit the projections (see [14]). 6.3

Typical results

The method we use now, needs to precise a priori the type of solution, we are looking for. This is a major difference with the case where (center manifold) reduction to an ODE is possible. For periodic solutions, we use an adaptation of Lyapunov-Schmidt method, except that the presence of 0 in the spectrum gives some trouble (resonant terms). It appears that we can formulate all these problems, such that there is no such resonant term. As a result, there are as many periodic solutions as in the truncated normal form (see [13]). For solutions homoclinic to 0 {solitary waves), we first inverse the infinite dimensional part of the system under normal form, using Fourier transform. Indeed, the linearized Fourier transform uses the above resolvent operator, where we eliminated, via a suitable projection, the poles given by eigenvalues sitting on the imaginary axis. The fact that the resolvent operator is not analytic near 0 (0 is not a pole, since we eliminated it, but there is still a jump of the resolvent in crossing the real axis [12]), leads to the fact that this "hyperbolic part" of the solution decays polynomially at infinity. The principal part of the solution at finite distance still comes from the finite dimensional truncated normal form, but its decays faster at infinity than the other part of the solution, which makes this queue part predominant at infinity. This is the main difference with the finite depth case, where the principal part coming from the normal form is valid for all values of x (see [12] for the proofs related with problem (1,2)).

92

G. looss

References [I] C.J.Amick, K.Kirchgassner. A theory of solitary waves in the presence of surface tension. Arch. Rat. Mech. Anal. 105, 1-49, 1989. [2] J.T.Beale. Exact solitary water waves with capillary ripples at infinity. Comm. Pure Appl. Math. 44, 211-257,1991. [3] F.Dias, G.Iooss. Capillary-gravity solitary waves with damped oscilla tions. Phys. D 65, 399-423, 1993. [4] F.Dias, G.Iooss. Capillary-gravity interfacial waves in infinite depth. Eur. J. Mech. B/Fluids 15, 3, 367-393, 1996. [5] G.Iooss. Capillary-Gravity water-waves problem as a dynamical system. Adv. Series in Nonlinear Dynamics 7, p.42-57, A.Mielke, K.Kirchgassner Eds., World Sci. 1995. [6] G.Iooss, M.Adelmeyer. Topics in Bifurcation theory and Applications. Adv. Series in Nonlinear Dynamics 3, World Sci. Pub. 1992. [7] G.Iooss. A codimension two bifurcation for reversible vector fields. Fields Institute Comm. 4, 201-217, 1995. [8] G.Iooss, J.Los. Bifurcation of spatially quasi-periodic solutions in hydrodynamic stability problems. Nonlinearity 3, 851-871, 1980. [9] G.Iooss, K.Kirchgassner. Water waves for small surface tension: an ap proach via normal form. Proc. Roy. Soc. Edinburgh 122A, 267-299, 1992. [10] G.Iooss, M.C.Pe>oueme. Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Diff. Equ. 102, 62-88, 1993. [II] G.Iooss. Existence d'orbites homoclines a un equilibre elliptique, pour un systeme reversible. C.R. Acad. Sci. Paris, Serie I 324, 993-997,1997. [12] G.Iooss, P.Kirrmann. Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves. Arch. Rat. Mech. Anal. 136, 1-19, 1996. [13] G.Iooss. Gravity and Capillary-Gravity periodic travelling waves for two superposed fluid layers, one being of infinite depth. J. Math. Fluid Mech. 1, 24-£l, 1999. [14] T.Kato. Perturbation theory for Linear Operators. Springer Verlag, 1966. [15] K.Kirchgassner. Wave solutions of reversible systems and applications. J. Diff. Equ. 45, 113-127, 1982. [16] T.Levi-Civita. Determination rigoureuse des ondes permanentes d'ampleur finie. Math. Annalen 93, 264-314, 1925.

The Water- Wave Problem as a Spatial Dynamical System 93

[17] E.Lombardi. Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves. Arch. Rat. Mech. Anal. 137, 227-304, 1997. [18] E. Lombardi Non-persistence of homoclinic connections for perturbed integrable reversible systems Journal of Dynamics and Differential Equa tions 11, 129-208, 1999. [19] A.Mielke. Reduction of quasilinear elliptic equations in cylindrical do mains with applications. Math. Meth. Appl. Sci. 10, 51-66, 1988. [20] S.M.Sun. Existence of generalized solitary wave solution for water with positive Bond number less than 1/3. J. Math. Anal. Appl. 156, 471-504, 1991. [21] S.M.Sun, M.C.Shen. Exponentially small estimate for the amplitude of capillary ripples of generalized solitary wave. J. Math. Anal. Appl. 172, 533-566, 1993. [22] A.Vanderbauwhede, CIooss. Center manifold theory in infinite dimen sions. Dynamics Reported 1 new series, 125-163,1992.

COLD ATOMS A N D MULTIPLE SCATTERING ROBIN KAISER Institut NonLiniaire de Nice In this article we use a classical description of laser cooling of atoms. In a second part we describe the use of cold atoms for multiple scattering experiments and discuss some effects which appear for dense atomic media.

Laser cooling of atoms is usually described using a quantized atomic sys tem (e.g. two-level system) and a classical description of the light. This is the so-called semi-classical description of laser cooling. It is also possible to quantize the light field, such as in the well known dressed states 1 . In this pa per however we want to use the opposite possibility and use the most classical description possible to understand laser cooling of atoms 2 . As we will see, it will not be possible to eliminate completely the quantum effects for laser cooling of atoms: the atomic resonance is known to be dependant on Planck's constant and the recoil of an atom after absorbing or emitting one photon is also quantized. But given these two points, it is possible to compute the basic effects of the so-called Doppler cooling by a classical model. If one accepts the resonance frequency as a input parameter and a phenomenological diffusion constant for the residual heating, one could even apply this model to other situation such as e.g. acoustic waves. However it seems difficult to use this scheme to efficiently cool other systems than individual atoms, even if it is worthwhile noting that solid samples of matter have now been cooled using laser light 3 . 1

Classical m o d e l of D o p p l e r cooling

Let us study the center of mass motion of atoms interacting with quasi res onant light. The radiative forces experienced by the atoms will depend on the detuning S between the laser frequency u>£ and the atomic resonant fre quency uiat. If for example one wants to compute the radiation pressure one needs to know the scattering cross section, which, in the case of particles with an internal resonance, can be much larger than the geometrical size of the particle. In order to take into account these internal resonance effects, we will model the atom as a kernel surrounded by an elastically bound electron, with a resonance frequency uat. The laser light drives the electron and thus induces a dipole d = qf = q (fe — K ) which, in the driven regime, oscillates 95

96 R. Kaiser

at the driving frequency u>£. It will be the interaction between this driven dipole and the electro-magnetic field of the laser which acts on the center of mass of the atom. We will hence proceed in two steps: first compute the dipole induced by the laser light and second study the motion of this oscillating dipole with the electromagnetic field. 1.1

Internal motion: elastically bound electron

In this model we will suppose that the distance f=fe — R between the electron and the kernel of the atom follows the equation 2

£f+r£?+ Wot 2 ?=J£

(i)

The total force acting on the electron is composed by the force fs due to the electric field at the position fe = ~R + r of the electron:

fs = qE(r„t) and by a component due to the magnetic field:

fB =

q^-At(fe,t)

The ratio between the amplitude of these two forces is of the order of

is. ~ *• I = us.«i fs di c c and we hence can neglect the effect of the magnetic field for computing the relative motion of the electron. Furthermore, the mass of the kernel being much larger than that of the electron, the distance r = re—R between the electron and the kernel of the atom is determined by the motion of the electron. We will use the complex notation for the electric field for a monochromatic linear polarized light: E(f, t) = E0(r)Z+ exp(-iw L t)

(2)

Using eqs. (1) and (2) and only taking into account the electric field force on gets a driven solution f(i) = f0 exp (—iwLt) with: -u)2Lr0-iuLTfo

+ uiat2f0 = 9 °™Zi

Defining the polarizability a(wi) of the atomic dipole by: d = q~?=e0

a(uL)E

Cold Atoms and Multiple Scattering

97

one thus obtains: a(wi)=

i

e

(w o t 2 -a;^-tw z ,r) e0me

We will use the real and the complex part of a(wi): a = a' + ia": a,

Uat2-u2L

_ 2

f_

2

2

Kt -w£) + (uar) «.»», „i and the atomic resonance frequency uat one gets for a quasi-resonant excitation (S « WL, WL — at) (figure 1):

98

R. Kaiser

- 6 - 4 - 2

0

2

4

frequency Figure 1. Atomic polarisability a[crou/{,/r]: real part a' (dashed line) and imaginary part a" (solid line) as a function of detuning u>j, — u/at[T]

a'

a"

_ —Sui

=71

(4)

..

(5)

~s'+4 2 _

$UL

"**+¥

2

Remarks i) In order to compute this polarizability we have approached the electric field Eo(fe) at the position of the electron with the field at R of the center of mass M of the atom. This approximation is valid if the distance between the electron and the center of mass of the atom is small compared to the scale on which the electric fielc varies, i.e. small compared to the wavelength of the laser: |f| = r e — R 4C A. This approximation is called electric dipole approximation as one can consider the atom as a point dipole on the scale of the wavelength A. Note that even in this dipole approximation, real atoms have a more complex internal structure (Zeeman sublevels e.g.) and exhibit some features which cannot be described by a classical dipole oscillation. ii) The damping of the dipole can be explained by the radiation of the oscil lating dipole. This radiation depends on the frequency of the oscillation and strictly speaking one should replace T by T ^ - . But we will only be interested


99

in frequencies close to the resonant frequency and we will thus neglect the change of the damping on the scale of 6. □ 1.2

Radiation forces acting on the atom: "Classical approach"

The force acting on the center of mass of atom, considered now as an oscillating dipole, has two components: one due to the electric field and one due to the magnetic field of the incident laser field, propagating along the Oz axes. The force f E due to the electric field, which we take polarized along the Ox axes:

E(r,t) =

E0(f)l}txp(-iuLt)

is directed parallel to this electric field:

'fE = J2qE(r,t)(x^x The magnetic force ~f B on the contrary will be directed along the axes of propagation of the laser (for a linear polarized plane wavej! The electron driven by the electric field has a velocity along the axes Ox: —£*■ = \^-^\ e% and for a magnetic field along Oy: B{~r*e,t) = Bo(~r*e)~e*y exp(-iwz,e,e B0(T*e)exp(-iwLt)-? dt

It is thus clear that one cannot neglect the effect of the magnetic field for computing the force acting on the center of mass of the atom! One can keep in mind the model of the electron driven by the electric field and it is the magnetic force which acts on this moving charged particle. Although this is not a complete description, it allows one e.g. to understand why the radiation pressure force is along the axes of propagation of the laser light, without using a quantum treatment for the electric field of the laser. Despite 4s- ~ | As d = q~~r* = e0 OI(UL)E

one can write:

n=jip*-*{%tj)-'tAjii[%t) Taking £~3(T$,t) ~ ■§i~^('^,t) (the velocities of the charges are small compared to c) and using Maxwell's equation ^- = — rot E one can express the magnetic force as a function of the electric field:

ft = jt p A #(!?,*)) + "^ A *#$ or, for the ■£? component:

/„ = i (V A*of,o)i+E ( one gets: aat = a"k, which depends on the detuning S (figure 2):

* * ( ' ) = „ ?i r » osL *

= {**)! T

The average time r between two spontaneous emissions is given by:

i = I^£

(12)

^

( r 2 I,at S2 + P / 4 ' This diffusion gives rise to an increase in the kinetic energy, i.e. a heating of the atoms. At equilibrium the heating due to the fluctuations and the cooling due to the friction effect compensate:

\2"

dJApf dt

= -21(Apfiq Uq

+ 2D = Q

(13)


0.0

-0.5 -1.0 -IS

-2.0 -IS

109

-34

detuning

Figure 4. Equilibrium temperature -gjh as a function of detuning S/T

and one obtains the temperature (figure 1.5): \2

L.T,M£L=.*.2 or other Rayleigh and Mie scatterers. 2.1

Scattering cross section of single atoms

Atoms can be considered as very strong scatterers with a cross section as large as 1 in Bose-Einstein condensates cold atoms, but localization effects in such samples have not yet been observed. One consequence of atoms closer than one wavelength is the correlation arising from recurrent scattering, which can also be interpreted as dipoledipole Van der Waals effects. For dilute sample, such that kIMFP >> 1 the correlation can lead to an increased "effective" mean free path (transport mean free path), as the radiation pattern can be affected to give enhanced forward scattering (superradiance e.g.). In dense media, the polarizability is modified

120 R. Kaiser due to local field effect (Lorentz-Lorenz formula 46,47 ) and we have X =

na 1-inor

or e-1 1 = -na e+2 3 with e = 1 + 4irxUsing a =

_1 op ■UJL -i\2

=

6 + i$ 2 a0 2 - 2

L

£ = 1 + Ann * - » E - £ >

= 1 + 4™L

which can be seen as Lorentz-Lorenz shift A A w

6-%fu,L

( ( J

AWLL

_

t t w L

+ iZ )

2 +

n

OCQ

2

Wi

of

n a

° " = -3TW

i

or using TT- = a o f c ^ or aow a( =f£yr one has (16)

AwLL = Using the relation 1

r = 3TT£

0

a;3 AC3

one can rewrite this shift hAwLL = —n

NT 3£n

This red-shift of the resonance is thus expected to be small for dilute samples, whereas for BEC samples e.g. one could expect shifts several times the natural line width of the transition. One can also write this result as: £ = 1 + Ann

uL = 1 + Ann — : - 5 — ^ —w/, (6 + Au>LL)2 + % 2


121

At resonance, for S = 0, one thus has w

W L ) 2, 2 + ^™ 2~ i>

e = 1 + 4™— which, for the absorptive part, is 4nn

^

op

2 (Aw t L ) + V and hence

2

*■=—=5- times smaller than without local field effects.

This

reduction factor

S

_

*

(17)

is equivalent to a reduced resonant cross section: Tret

1

=

(IS) +1 The "on-resonant" mean free path thus is increased:

^((SH-**

3A 3 2ir

and one obtains the Joffe-Regel criterion (figure 8)

(tf)' + . .(£)'+. r

" ~

„3A3 "4^

~" Q d

/nA»\ (.4^)

which is minimal for ^£y = 1 and is then ~

t;

2

< 1

" " 3 " This model seems to predict that strong localization of light in dense cold atomic vapors can only be obtained for on-resonant excitation in a narrow window of density. One has however to investigate more precisely how the fluctuations of such a "mean field effect" will modify not only the position of the resonance but also its width. The bare shift of the resonance could be taken care of by taking e.g.: 6 + AwLi = 0

122

R.

Kaiser

s u an o Oi

&o

Figure 8. Joffe-Regel parameter •> as a function of density nA 3 /4ir 2

One then has:

=

^((r--4^j +1J

and hence the detuning dependant Joffe-Regel parameter (figure 9):

a

i (»-£)'+' »

(3U)

This should compensate for this Lorentz-Lorenz shift and it should then be possible to keep the same cross section as without local field effects. In that case the Joffe-Regel criterion seems to be fulfilled when 1 1

[*!,„ = i < ■ 4»3

i.e. when nX3>^-

4TT 2

(18)

Cold Atoms and Multiple Scattering 123

4

3

2

1

0 -4

-2 0 2 4 6 detuning

8

10

Figure 9 . Joffe-Regel parameter y as a function of detuning 8/F

Note that in this result, only the optical wavelength is relevant (as opposed to the threshold for Bose-Einstein condensation where one requires nA B > 2.613 and where the De Broglie wavelength is the important parameter) The Lorentz-Lorenz correction has been described in more detail for optical thin media by Friedberg et al46. The precise shift depends on geometrical configuration and give different numerical factors. It is interesting to notice that even the anti-resonant terms (usually neglected in teh rotating wave approximation) contributes significantly to the shift. Furthermore collisional shift of the resonance cannot be neglected as it also varies linearly in density (as is well known in atomic clocks): OGlcoll =

On

This effect is difficult to distinguish from the Lorentz-Lorenz correction as both scale as the density of the atoms . However in a non linear experiment with hot atoms it has been shown that the Lorentz-Lorenz correction has to be taken into account for a precise evaluation of the shift of the atomic resonance in dense media. Furthermore the Doppler effect results in an inhomogeneous broadening of the line and the "effective" average cross section for resonant

124 R. Kaiser

detuning will be reduced. As the temperature dependance of the atoms does not appear explicitly in eq.(18) it is important to note that in addition to the shift of the resonance collisional broadening of the line, which also scales as the density of the atomic gas, cannot be neglected. 3

Conclusion

In this paper we have described the basic cooling mechanism of atoms by laser using a (almost) complete classical description. This approach can be very usefull to estimate the cooling (in-)efficiency in other situations (dielectric spheres with internal resonances e.g.). In the second part we have presented a well known feature of interference effect in multiple scattering which we have applied to atoms as scatterer. Extended basic criteria to high density, we have addressed the question of how the Joffe-Regel criterion is modified and where one can expect strong localization of light in dense atomic samples. This very interesting regime however needs more thorough investigation. We think that multiple scattering in dense cold atomic media is a very promising topic which is now accessible with e.g. Bose-Einstein condensates. References 1. C.Cohen-Tannoudji, J.Dupont-Roc, G.Grynberg, Processus d'interaction entre photons et atomes, Intereditions, Paris, 1988. 2. G.Grynberg, A.Aspect, C.Fabre, Introduction aux laser et a I'optique quantique, Ellipses, 1997. 3. C. E. Mungan, M. I. Buchwald, B. C. Edwards, R. I. Epstein, and T. R. Gosnell, Phys. Rev. Lett. 78, 1030 (1997). 4. C.Cohen-Tannoudji, College de France lectures 1982. 5. T.W.Hansch, A.Schawlow, Opt. Comm. 13, 68 (1975). 6. D.Wineland, H.Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975). 7. W.D.Phillips, Rev. Mod. Phys. 70, 721 (1998). 8. S.Chu, Rev. Mod. Phys. 70, 685 (1998). 9. C.Cohen-Tannoudji, Rev. Mod. Phys. 70, 707 (1998). 10. J. Dalibard, C. Cohen-Tannoudji, J.O.S.A.z B 6 , 2023 (1989). 11. A.Aspect, E.Arimondo, R.Kaiser, N.Vansteenkiste, C.Cohen-Tannoudji, Rev. Mod. Phys. 6 1 , 826-829 (1988). 12. V.B.Berestetskii, E.M.Lifshitz, L.P.Pitaevskii, Quantum Electrodynam ics, 2nd edition 1982 (course of theoretical physics vol 4), (ButterworthHeinemann ed., Reed Elsevier, ISBN 0 7506 3371 9) p.246.

Cold Atoms and Multiple Scattering 125

13. H.Mabuchi, H.J.Kimble, Opt. Lett. 19, 749 (1994). 14. J.CKnight, N.Dubreuil, V.Sandoghdar, J.Hare, V.Lefevre-Seguin, J.M.Raimond, S.Haroche, Opt. Lett. 20, 1515 (1995). 15. A.Lagendijk, B.v.Tiggelen, Phys. Rep. 270,143 (1996). 16. G.L.Lippi, G.P.Barozzi, S.Barbay, J.R.Tredicce, Phys. Rev. Lett. 76, 2452 (1996). 17. L.Mandel, E.Wolf Optical Coherence and Quantum Optics, Cambridge University Press (1995). 18. A.E.Siegman, "Lasers", University Science Books (1986). 19. M.Fleischhauer, S.Yelin, Phys. Rev. A59, 2427 (1999) 20. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269, 198 (1995). 21. http://amo.phy.gasou.edu:80/bec.html/ 22. S.Inouye, A.P.Chikkatur, D.M.Stamper-Kurn, J.Stengr, D.E.Pritchard, W.Ketterle, Science 285, 571 (1999). 23. A.Steinberg, P.G.Kwiat, R.Ciao, Phys. Rev. Lett. Lett. 71 , 708 (1993). 24. H.M.Brodwosky, W.Heitman, G.Nimtz, Phys. Rev. A 222 , 125 (1996). 25. E.H.Hauge, J.A.Stovneng, Rev. Mod. Phys. 6 1 , 917 (1984). 26. V.S.Olkhovsky, E.Recami, Phys. Rep. 214, 339 (1992). 27. New Aspects of Electromagnetic and Acoustic Wave Diffusion, Springer Tracts in Modern Physics, vol.144, POAN Research group (ed.), 1998. 28. Laser Speckle and Related Phenomena, Topes in Applied Physics, vol 9, J.C.Dainty (ed. )Springer-Verlag (1975). 29. This assumption will break down in the case of a Faraday eifect or in the case of moving scatterers, leading to a decrease of the coherent backscattering cone. 30. Y. Kuga and A. Ishimaru, J. Opt. Soc. Am. A l , 831 (1984). 31. RE. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985). 32. M.P. van Albada, A.Lagendijk, Phys. Rev. Lett. 55 , 2692 (1985). 33. D. Wiersma, M. van Albada, B. van Tiggelen and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995). 34. A.F.Joffe, A.R.Regel, Progress in semiconductors 4, 237 (1960). 35. G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Miiller, C Miniatura and R. Kaiser, Phys. Rev. Lett. 83, 5266 (1999). 36. R.H.Dicke, Phys. Rev. 93, 99 (1954). 37. M.Gross, S.Haroche, Phys. Rep. 93, 301 (1982). 38. D.Polder, M.F.H.Schuurmans.Pht/s. Rev. A14, 1468 (1976). 39. C.Cohen-Tannoudji, S.Reynaud, J. Phys. 38, L173 (1977). 40. C.Cohen-Tannoudji, S.Reynaud, J. Phys. B 10, 365 (1977). 41. B.R.Mollow, Phys. Rev. 188, 1969 (1969). B.R.Mollow, Progress in

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Optics, vol XIX, 1, (E.Wolf ed.) North Holland (1981). 42. G.Birkl, M.Gatzke, I.H.Deutsch, S.L.Rolston, W.D.Phillips, Phys. Rev. Lett. 75, 2823 (1995). 43. M.Weidemiiller, A.Hemmerich, A.Gorlitz, T.Easlinger, T.W.Hansch, Phys. Rev. Lett. 75, 4583 (1995). 44. C. Jurczak, K. Sengstock, R. Kaiser, N. Vansteenkiste, C.I. Westbrook, A. Aspect, Opt. Commun. 115, 480 (1995). 45. D.Wiersma, P.Bartolini, A.Lagendijk, R.Righini, Nature 390, 671 (1997). 46. R.Friedberg, S.R.Hartmann, J.T.Manassah, Phys. Rep. 7, 101 (1973). 47. Y.Castin, K.Moellmer, Phys. Rev. A51, R3426 (1995).

A N INTRODUCTION TO ZAKHAROV THEORY OF WEAK TURBULENCE MICHEL LE BELLAC Imtitut Mm Liniairt de Nice These lectures give a (hopefully!) pedagogical introduction to the theory of non linear interactions of waves and weak turbulence. After having derived the pertubative expansion of the Hamiltonian of surface waves, we write down the kinetic equations for the number density of waves. Then stationary solutions to these kinetic equations allow us to find the weak turbulence spectrum.

1

Introduction

These lectures address the problems of nonlinear interaction of waves and of weak turbulence. Actually, we shall deal mostly with the problem of surface waves, but many of the concepts which are introduced in these lectures also apply to other kinds of wave: sound waves, spin waves, Langmuir waves in plasmas, Rossby waves in the atmosphere etc. In introductory textbooks, waves are usually treated in the linear (or, equivalently, harmonic) approxi mation: the simplest example is probably wave propagation in an harmonic solid. An elementary one-dimensional model is as follows: one considers a set of N identical masses m vibrating on a line, each mass being linked to its two neighbours by identical springs (Fig. 1). The equilibrium position of the tth mass is denoted by xk is the dispersion law . \ka n [K — (1.4) w* = 2\V —sin m 12 where a is the lattice spacing. When ka +i(v

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Nonlinear Phenomena in Power Electronics:attractors,bifurcations,chaos,and nonlinear control

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Questions on Algebraic Varieties (C.I.M.E. Summer Schools, 51)

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Summer on the Mountain

Summer on the Mountain

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Materials with Memory (C.I.M.E. Summer Schools, 74)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Stability Problems (C.I.M.E. Summer Schools, 65)

Peyresq lectures on nonlinear phenomena : [lectures of two summer schools, the INLN Summer Schools on Nonlinear Phenomena, in June 1998 and June 1999]

Nonlinear optimization: Lectures C.I.M.E. Summer School, 2007

Lectures on Nonlinear Dispersive Equations

Lectures on Nonlinear Dispersive Equations

Lectures on phase transitions and critical phenomena

Lectures on nonlinear hyperbolic differential equations

Lectures on Nonlinear Hyperbolic Differential Equations

Nonlinear Phenomena in Power Electronics:attractors,bifurcations,chaos,and nonlinear control

Nonlinear Phenomena in Power Electronics:attractors,bifurcations,chaos,and nonlinear control

Stereodynamics (C.I.M.E. Summer Schools, 56)

Nonlinear Dielectric Phenomena in Complex Liquids

Nonlinear Singular Perturbation Phenomena: Theory and Applications

Controllability and Observability (C.I.M.E. Summer Schools, 46)

Questions on Algebraic Varieties (C.I.M.E. Summer Schools, 51)

Differential Operators on Manifolds (C.I.M.E. Summer Schools, 70)

Progress in nonlinear analysis : proceedings of the Second International Conference on Nonlinear Analysis, Tianjin, China, 14-19 June 1999

Summer on the Mountain

Summer on the Mountain

Stochastic Differential Equations (C.I.M.E. Summer Schools, 77)

Dynamical Systems (C.I.M.E. Summer Schools, 78)

Wave Propagation (C.I.M.E. Summer Schools, 81)

Wave Propagation: (C.I.M.E. Summer Schools, Vol. 81)

Teoria Della Turbolenza (C.I.M.E. Summer Schools, 14)

Differential Topology (C.I.M.E. Summer Schools, 73)

Algebraic Surfaces (C.I.M.E. Summer Schools, 76)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Complex Analysis (C.I.M.E. Summer Schools, 62)

Materials with Memory (C.I.M.E. Summer Schools, 74)

Potential Theory (C.I.M.E. Summer Schools, 49)

Spectral Analysis (C.I.M.E. Summer Schools, 64)

Stability Problems (C.I.M.E. Summer Schools, 65)

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