W.-H. Steeb was born in Pforzheim, Germany, in 1945 and was educated at the University of Kiel. He received his Ph.D. fr...
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W.-H. Steeb was born in Pforzheim, Germany, in 1945 and was educated at the University of Kiel. He received his Ph.D. from the University of Kiel in 1976 and his post-doctoral degree (Dr. habil.) from the University of Paderborn in 1981. Presently, he is Professor of Theoretical Physics and Applied Mathemat ics in the Department of Applied Mathematics and Nonlinear Studies, Rand Afrikaans University, Johannesburg, South Africa. His research interests are in the areas of statistical physics and nonlinear dynamics, in particular chaos with applications in electronics, quantum chaos and integrable systems. He is also involved in work on applications of computer algebra in science. He has more than 140 research papers and 17 books to his credit.
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WERTIILE POINT TRANSFORMATIONS AND NONLINEAR DIFFERENTIAL EQUATIONS Willi-Hans Steeb Rand Afrikaans University
World Scientific Singapore • •New Singapore NewJersey Jersey• L• London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
INVERTIBLE POINT TRANSFORMATIONS AND NONLINEAR DIFFERENTIAL EQUATIONS Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyany means, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN 981-02-1355-7
Printed in Singapore.
Preface For nonlinear ordinary and partial differential equations the general solution usually cannot be given explicitly. It is desirable to have an approach by which it can be de termined whether a given nonlinear differential equation is integrable. Several methods have been employed for studying the existence of first integrals and the integrability of dynamical systems. A powerful tool to find integrable differential equations (both ordi nary or partial) is the Painleve test (Steeb and Euler 1988). For classical Hamiltonian systems we can apply the Ziglin analysis or the method of Noether symmetries (Eu ler and Steeb 1992). Other methods can also be applied to non-Hamiltonian systems: t h e direct method, the linear compatibility analysis, the use of Lax pairs, t h e method of Lie symmetries, t h e quasimonomial formalism, and the Carleman embedding proce dure (Kowalski and Steeb 1991). In this book we describe the method of t h e invertible point transformation. We also discuss the connection with other methods mentioned above. T h e invertible point transformation allows the construction of nonlinear differ ential equations from linear differential equations. In chapter 1 first-order ordinary differential equations and the invertible point transfor mation is studied. As an example we consider the Bernoulli equation. In chapter 2 we investigate second-order ordinary differential equations and t h e invertible point transformation. We also consider systems of second-order ordinary differential equations. A large number of examples illustrate the approach. T h e investigation of third-order differential equations is not very common in classical mechanics due to t h e specific form of the fundamental newtonian equations. These kind of equations appear in modelling of physical situations such as a radiating charged parti cle interacting with an external electromagnetic field. They appear also in the reduction procedure of nonlinear partial differential equations by similarity transformations (for example the Korteweg de Vries equation). In chapter 3 we discuss third-order ordinary differential equations and the invertible point transformation. T h e m e t h o d introduced by Lie considers t h e invariance of the form of the differential equation itself under invertible point transformations of one parameter. Lie himself showed t h a t for t h e one-dimensional free particle there are eight point transformations of one p a r a m e t e r t h a t maintain the invariance of t h e equation; the same situation occurs for a time-dependent oscillator. This is t h e maximum number of generators for a secondorder differential equation of the form cPu/dt2 = H(du/dt,u(t),t). Certain ordinary nonlinear differential equations also have eight symmetry generators. T h e free particle
v
equation d2U/dT2 = 0 has eight Lie symmetry vector fields. We generate a class of nonlinear second order differential equations with eight Lie symmetry vector fields by applying an invertible point transformation to the free-particle equation. We also show that this transformation permits us to obtain directly the symmetry generators for this class of equations by using the symmetry generators of the free particle. As examples we consider: the harmonic oscillator, the time-dependent oscillator, the Kepler problem, the particle in a constant magnetic field, and the chargemonopole interaction. In chapter 4 the connection between Lie point symmetries and the invertible point transformation is studied. In chapter 5 the connection between first integrals and the invertible point transforma tion is discussed. A particular problem is the identification of the classes of linearizable equations. For the second-order differential equations this amounts to finding the class of equations which are equivalent to the free-particle equation. The Cartan equivalence method provides the answer to this problem. We also consider third-order ordinary differential equations. In chapter 6 the Cartan method is introduced. In chapter 7 we study the invertible point transformation and the Painleve test. As an application we consider the anharmonic oscillator and the second Painleve transcen dents. For ordinary differential equations the Painleve analysis and the invertible point transformation can be used to construct integrable nonlinear equations or equations which are related to the Painleve transcendents. We compare both techniques for the second Painleve transcendents. First we give an introduction to the Painleve test and then consider its connection with the invertible point transformation. In chapter 8 we consider the Painleve test and partial differential equations. In chapter 9 partial differential equations and the invertible point transformation is discussed. We also study its connection with the Painleve test. In chapter 10 we consider difference equations and linearization. As an example the logistic equation is studied. Most of the calculations are checked with REDUCE. A collection of programs which are helpful in the study of the invertible point transformation is given in chapter 11. Chapter 12 gives a short introduction to the jet bundle formalism.
vi
Contents 1 First-order ordinary differential equation
1
2 Second-order ordinary differential equations 2.1 Second-order ordinary differential equation 2.2 Systems of second-order differential equations
7 7 27
3 Third-order differential equations
35
4 Lie point symmetries
45
5 First integrals and differential equation
61
6 Cartan equivalence method 6.1 Second-order differential equations
69 69
6.2 Third-order differential equations
76
7 Painleve test and linearization
87
8 Painleve test and partial differential equations
105
9 Partial differential equations
117
10 Difference equations
133
11 R E D U C E programs
141
12 Jet bundle formalism
171
References
175
Index
179
vii
Symbol Index
empty set the set of integers the set of positive integers: natural numbers the set of rational numbers the set of real numbers nonnegative real numbers the set of complex numbers the n-dimensional real linear space the n-dimensional complex linear space
0
z M Q 11 11+ C TV C" i %z AcB AnB AUB f°9 u,U t,T x,X u = U =
z II. II xy det / [,] Sjk
A x A d(.)
c
(u1,u2,...,un)T (UuU2,...,Unf
real part of the complex number z imaginary part of the complex number z the element x of Hn the subset A of the set B the intersection of the sets A and B the union of the sets A and B composition of mappings ( / o g)(x) = f(g{x)) dependent variables independent variables (time variables) independent variables (space variables) vector of dependent variables vector of dependent variables Lie symmetry vector field norm scalar product (inner product) determinant unit matrix, identity operator commutator Kronecker delta with Sjk — 1 for j = h and Sjk = 0 for j ■£ k eigenvalue cross product Grassmann product (exterior product, wedge product) exterior derivative Lagrangian function viii
Chapter 1 First-order ordinary differential equation An ordinary differential equation of first-order
4Gfa 5G' du dt dt
(10)
Let us now give several examples: Example 1: The Bernoulli equation dti
Po(t)-£ + Pi(t)u =
fl(t)u"
(n#l)
(11)
is transformed by the invertible point transformtion T(t) = t,
U(T(t)) = u*-»(t)
(12)
into the linear differential equation + (1 - n)P1(T)U = (1 - n)R(T).
Po(T)^
This can be seen as follows: Since
_ dT
(14)
Hi and
dU
dUdT
(13)
.,
-dT = Ir!t={1-n)u
,
_ndu
Tt
( 15 )
3 we have dU dll
„„du du
,
1T = ^~-n)u-*-.
a
(16)
Example 2: The homogeneous differential equation
du du
H
+ f
.(u\ fu\
{l)
(17)
=0
is transformed by
T(t) = t,
U(T(t)) =
^l
(18)
into the separable equation dU U + T— + f(U) = Q.
(19)
In examples 1 and 2 only the dependent variable is changed. D Example 3: The differential equation du dt
/
I at + bu + c \ at + 0V. + 7
(20)
may always be integrated by an invertible point transformation Here a, 6, c, a, ft, 7 are real constants. There are two cases to consider: Case 1:
D
(a
bx
-**{l u 0,) , o . "/,)**■
(21)
We make the invertible point transformation
t(T)=T + h,
u(t(T)) = U(T) + k
(22)
where h and k are constants. It follows that du dU ~dt ~~ dT
(23)
aT + bU + ah + bk + c aT + 0U + ah + (3k + 7
(24)
and at + bu + c at + j3u + f Since D ^ 0, the linear equations ah + bk + c = 0,
ah + /3k + 7 = 0
(25)
4
CHAPTER 1. FIRST ORDER ORDINARY DIFFERENTIAL
EQUATION
may be solved uniquely for h, k. With these values, (20) becomes dU dT
(aT + bU\ \aT + pU)'
J
(26)
This equation is homogeneous and may be integrated. Equations (22) may be interpreted as a translation of rectangular axes to the new origin (h, k), the point of intersection of the lines at + bu + c = 0,
at + 0u + 7 - 0.
(27)
When these lines are parallel, we have the second case: Case 2: D = det(*
>)=0.
(28)
We now put
T(t) = t,
U(T(t)) = at + bu® = !*±m.
(29)
These fractions are equal since D = 0. Now dU_ = 1+ ~ dT~ ' adt dU
(30)
and (20) is reduced to dU b (aU + =1+ dT = a \aU + in which the variables are separable. Example 4-' Let us consider du dt
□
i)
Zt — u — 5 -t + 3u + T
(31)
(32)
The lines 3t - u - 5 = 0,
-t + 3u + 7 = 0
(33)
meet in the point (1, —2). The change of variable t{T) = T + 1,
u(t(T)) = U(T) - 2
(34)
reduces (32) to the homogeneous equation dU _ 3T-U dT~ -T + 3U
(35)
5 or
(3T - U)dT + (T - 3U)dU = 0.
(36)
Since this admits the multiplier //. 1 T{3T -U) -- U) + U{T - == T(3T U(T --3*7) - W) = 3(T 2 - U2) the equation
3T- -u
T- -3U d,rr* » =n
^
d,T ++ ^
rpi _ ■Ul
(38)
JXk kt - 1C tUc kt 2+ 6C u(t) =
+ C2.
2
x
where C\ and C? are constants of integration.
2
(38)
(39) ™
□
Example 5: Consider the differential equation for the free-falling body «PH
eft2
(40)
+