Symmetries of Partial Differential Equations: Conservation Laws — Applications — Algorithms

453 Pages·1990·31.04 MB·English

by  A. M. Vinogradov (auth.), A. M. Vinogradov (eds.)

SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS
Symmetries of
Partial Differential
Equations
Conservation Laws - Applications - Algorithms
Edited by
A.M. VINOGRADOV
Moscow Stale University, U.S.S.R.
Reprinted from Acta Applicandae Mathematicae, Vol. 15, Nos. 1 & 2 and
Vol. 16, Nos. 1 & 2
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Symmetries of partial differential equations conservation laws.
applications. algorithms / edited by A.M. Vinogradov.
p. em.
Translated from Russian.
"Reprinted from Acta applicandae mathematicae. volume 15. no. 1-2
and volume 16. no. 1-2."
1. Differential equations. Partial--Numerical solutions.
I. Vinogradov. A. M. (Aleksandr Mikha,lovich) II. Acta appllcandae
mathematicae.
QA377.S963 1990
515' .353--dc20 89-26687
ISBN-13: 978-94-010-7370-7 e-ISBN-13: 978-94-009-1948-8
DOl: 10.1007/978-94-009-1948-8
Published by Kluwer Academic Publishers.
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All Rights Reserved
© 1989 by Kluwer Academic Publishers, Dordrecht, The Netherlands
Softcover reprint of the hardcover 1s t edition 1989
No part of the material protected by this copyright notice may be reproduced or utilized in
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Table of Contents
PART I (Acta Appl. Math. 15,1-210)
A. M. VINOGRADOV / Foreword
A. M. VINOGRADOV / Symmetries and Conservation Laws of Partial Differential
Equations: Basic Notions and Results 3
V. N. GUSYATNIKOVA, A. V. SAMOKHIN, V. S. TITOV, A. M.
VINOGRADOV, and V. A. YUMAGUZHIN / Symmetries and Conservation
Laws of Kadomtsev-Pogutse Equations (Their computation and first
applications) 23
V. N. GUSYATNIKOVA and V. A. YUMAGUZHIN / Symmetries and Conserva-
tion Laws of Navier-Stokes Equations 65
N. O. SHAROMET / Symmetries, Invariant Solutions and Conservation Laws of
the Nonlinear Acoustics Equation 83
A. M. VERBOVETSKY / Local Nonintegrability of Long-Short Wave Interaction
Equations 121
V. S. TITOV / On Symmetries and Conservation Laws of the Equations of Shallow
Water with an Axisymmetric Profile of Bottom 137
YU. R. ROMANOVSKY / On Symmetries of the Heat Equation 149
I. S. KRASIL'SHCHIK and A. M. VINOGRADOV / Nonlocal Trends in the
Geometry of Differential Equations: Symmetries, Conservation Laws, and
Backlund Transformations 161
PART II (Acta Appl. Math. 16, 1-142)
A. N. LEZNOV and M. V. SA VELIEV / Exactly and Completely Integrable
Nonlinear Dynamical Systems
B. G. KONOPELCHENKO / Recursion and Group Structures of Soliton Equations 75
V. O. BYTEV / Building of Mathematical Models of Continuum Media on the
Basis of the Invariance Principle 117
TABLE OF CONTENTS vi
PART III (Acta Appl. Math. 16,143-242)
A. V. BOCHAROV and M. L. BRONSlEIN / Efficiently Implementing Two
Methods of the Geometrical Theory of Differential Equations: An Experience
in Algorithm and Software Design 143
E. V. PANKRAT'EV / Computations in Differential and Difference Modules 167
V. L. TOPUNOV / Reducing Systems of Linear Differential Equations to a Passive
Form 191
PAlJL H. M. KERSTEN / Software to Compute Infinitesimal Symmetries of
Exterior Differential Systems, with Applications 207
P. K. H. GRAGERT / Lie Algebra Computations 231
Acta Applicandae Mathematicae 15: 1-2, 1989.
© I <i89 Kluwer Academic Publishers.
Foreword
This special double issue of Acta Applicandae Mathematicae and two further
issues to appear in the future, are devoted to recent developments in the theory
of symmetries and conservation laws for general systems of partial differential
equations.
The first part deals with the problem of how to find all higher symmetries and
conservation laws for a given differential equation. It opens with an article in
which the necessary theoretic results are summarized and which is the starting
point for the subsequent articles. The above-stated problem is partially or
completely solved for concrete equations of mathematical physics which are
chosen to illustrate, from different points of view, both the techniques of
computations as well as the characters of the problems which arise. Some
attention is also paid to applications. These are mainly related to invariant
solutions. The first steps toward the general non local theory of symmetries and
conservation laws are made in the final paper.
The following two points are to be stressed. Firstly, equations with more than
two independent variables are mainly investigated here and, for some of these, all
higher symmetries and conservation laws are computed completely. Up to now,
similar strong results were obtained only for equations with two independent
variables. Secondly, the presented method of generating functions for finding
conservation laws does not depend on any symmetry considerations. In parti-
cular, it works effectively in situations where Noether-type theorems are not
applicable.
The Leznov and Saveliev and Konopel'chenko papers in the second part of this
special volume deal with some constructions of 'integrable' (in a sense) systems of
nonlinear differential equations. Many interesting properties of these systems
arising from their nature, are found here as well as explicit formulae for some
wide classes of their solutions. In other words, some relations between two
conceptions of 'integrability' and 'symmetry' are investigated in these papers. In
the final paper of the second part, written by O. Byter, some methods for
modelling continuum media based on symmetry considerations are proposed.
It is now obvious that effective applications of symmetry methods to in-
vestigate concrete equation are impossible without considerable computer sup-
port. For this reason, the third part of these special issues concerns these topics.
Some recent trends along this line are presented in it. However, this issue is far
from a full account of modern activities in this field.
In this volume, the editor has tried to reflect all the main recent results and
tendencies in the considered domain. Of course, such a goal cannot be reached,
but I hope that the reader will find here a satisfactory approximation of it.
2 FOREWORD
The authors of these issues involve not only mathematicians, but also speci-
alists in (mathematical) physics and computer sciences. So here the reader will
find different points of view and approaches to the considered field.
A. M. VINOGRADOV
Acta Applicandae Mathematicae 15: 3-21, 1989. 3
© 1989 Kluwer Academic Publishers.
Symmetries and Conservation Laws
of Partial Differential Equations: Basic
Notions and Results
A. M. VINOORADOV
Department of Mathematics, Moscow State University, 117234, Moscow, U.S.S.R.
(Received: 22 August 1988)
Abstract. The main notions and results which are necessary for finding higher symmetries and
conservation laws for general systems of partial differential equations are given. These constitute the
starting point for the subsequent papers of this volume. Some problems are also discussed.
AMS subject classifications (1980). 35A30, 58005, 58035, 58H05.
Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged
equations, generating functions.
o. Introduction
In this paper we present the basic notions and results from the general theory of
local symmetries and conservation laws of partial differential equations. More
exactly, we will focus our attention on the main conceptual points as well as on
the problem of how to find all higher symmetries and conservation laws for a
given system of partial differential equations. Also, some general views and
perspectives will be discussed. The material of this paper is used in other papers
[7]-[12] of this volume in which the general theory is applied to concrete
equations of mathematical physics. This demonstrates the theory in action.
Presented here are the results on higher symmetries and conservation laws
found by the author in 1975-1977 and its resume was published in a short note
[1] (see also [2]). However, the author was not successful in publishing full details
until 1984. Between these years, other authors developed similar ideas (Ibragi-
mov [5], Olver [6] etc.) in the field of symmetry theory. Tsujishita's work [17]
contains some results on conservation laws which are close to ours. Higher
symmetries and conservation laws for spatially one-dimensional evolution equa-
tions have been the subject of many works on the investigation of equations
integrable by the inverse scattering transform method. We do not touch on these
very interesting but very special topics in this paper.
In further exposition we will use the usual coordinate language for mathemati-
cal physics. However, this is not the best way to think of these substances. We
omit here both motivations of basic notions (these are given in [3]) and proofs or
4 A. M. VINOGRADOV
their indications for the presented results. An interested reader will cover this gap
by consulting [4,13,17]. We also recommend paper [14] in which all the main
technological details necessary for finding symmetries and conservation laws are
demonstrated in a concrete example.
Probably the most interesting new point of the theory presented below is that in
principle, it makes it possible to find all higher conservation laws for arbitrary
(nonlinear) differential equations. In particular, it works effectively well in
situations when the Nother theorem, as well as other symmetry considerations,
are not applicable. How it looks from the practical point of view will be clear
from subsequent papers.
1. On Terminology
Below, the theory of higher local infinitesimal symmetries and conservation laws
of partial differential equations is discussed. We use the adjective 'higher' to
stress that the symmetries and conservation laws under consideration are de-
scribed by means of expressions containing arbitrary order derivatives of quan-
tities, entering into investigating differential equations.
The adjective 'local' is used to point out that we deal with symmetries and
conservation laws which admit localizations on arbitrary domains in the space of
independent variables. Foundations of nonlocal theory are considered in [15] in
this volume.
The classical symmetries theory, originated by S. Lie, operates, with first-order
derivatives. By speaking of 'higher symmetries', we underline the aspect which
differentiates the modern theory from the classical one. Some authors use
'generalized symmetries' or 'Lie-Backlund transformations' in the same sense.
The last term seems to be very misleading because the notion of 'Backlund
transformation' is a concept of a quite different nature. In particular, higher
symmetries are infinitesimal transformations, but Backlund transformations are
finite ones.
Below, using the word 'symmetry', we have in mind 'higher local symmetry'.
2. Infinitely Prolonged Equations
Informally, infinitesimal symmetries are infinitesimal transformations of manifolds
of infinitely prolonged equations which conserve their natural contact structures.
For this reason, we consider these notions in more detail.
Let x = (XI, ... , xn) be independent variables and u = (u l , ... , urn) be depen-
dent ones. Geometrically, this means that we deal with a smooth fibre bundle
1T: E ~ M, x's are the base coordinates in it and u's are the fibre coordinates.
In some situations below, the multiindexes
u = (iI, ... ,in), i = 1, 2, ... , n,
SYMMETRIES AND CONSERVATION LAWS OF PDEs 5
will be written in the form
u= 1 ... 1, 2 ... 2, ... , n ... n.
~ ~ '--'
it times i2 times in times
Also, we will use the following short notation
iJl u I iJ i, +- - -+in
iJxu axi' ... iJx~'
supposing that u = (il , ... , in) and Iu I = il + ... + in .
Now let us introduce the variables p~, 1 ~ i ~ m, 'flu. The manifold with local
coordinates x, u, p~, 1 ~ Iu l ~ k, 1 ~ i ~ m, is said to be the manifolds of the kth
order jets of 7T, or simply the kth jet manifold. More exactly, variables
x, u, ... ,p~, ... are local coordinates on the manifold J k7T of kth-order jets
associated with the fibre bundle 7T. If k < 00, then J k 7T is a finite-dimensional
manifold. However, infinite-dimensional manifolds Joo = Joo 7T will be of most
interest to us. The local coordinates on Joo are x, u, p~, where 1 ~ i ~ m,
Iu I < 00. A smooth function on J 00 is, by definition, a smooth function depending
only on a finite number of variables x, u, p~. The entity of all smooth functions
on Joo will be denoted by 8fr= 8fr(7T). The part of 8fr consisting of all functions
depending only on variables x, u, p~, with Iu l ~ k, will be denoted by 8frk =
8frk ( 7T). Evidently
Below, we will write pu instead of p~ in the case m = 1 and will sometimes use
the notation p~ instead of ui •
It will be convenient for us to trait a kth-order system of partial differential
equation (PDE)
(1)
(Here, U(s) denotes the totality of all derivatives iJlului/iJxu, lui = k) as a sub-
manifold cy in Jk which is given by the equations
FI(X,u, ... ,p:;,., ... )=O} lul~k.
(I')
FI (x, U, ... , p:;", ...) = 0
For example, from this point of view, the wave equation looks like the
hyperplane in the space of variables Xl = x, X2 = t, U, PI, P2, Pll, P12, P22 whose
equation is Ptt - P22 = 0 (or P(2.0) - P(O,2) = 0 if one uses the standard notations for
multi-indexes.)
The full derivative operator with respect to Xk D k : 8fr ~ 8fr is defined by the