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 DOl: 10.1007/978-94-009-1948-8
e-ISBN-13: 978-94-009-1948-8
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Table of Contents
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
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. © 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)
3
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
will be written in the form
u= 1 ... 1, ~
iJl u I iJ i, +- - -+in
iJxu axi' ... iJx~'
supposing that u = (il , ... , in) and I u I = il + ... + in
.
5
Now let us introduce the variables p~, 1 ~ i ~ m, 'flu. The
manifold with local coordinates x, u, p~, 1 ~ I ul ~ 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 k 7T 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, I u 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 I ul ~ 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 u i •
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. FI (x, U, ... , p:;", ... ) =
0
(I')
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
6 A. M. VINOGRADOV
(JXk u,s (Jp"
Here ak = (iI, ... , ik + 1, ... , in), supposing that a = (iI, ...
, in). If m = 1, then
Let
DO' = D\. 0··· 0 D~
if a = (it , ... , in). Evidently, D;Dj = Dp; . The submanifold Woo
in Joo which is given by the infinite system of PDE,
Du(F;) = 0, Va, i (2)
is called the infinite prolongation of the system W defined by (1).
Generally, Woo is an infinite-dimensional manifold.
Local coordinates x, U, p~ on Joo being restricted on Woo, are not
yet independent. In fact, some corrdinates can be expressed through
others by using Equation (2). In other words, these coordinates may
be divided into two parts: internal and external, with respect to
Woo. More exactly, the maximal functionally independent part of
coordinates x, U, p~ on Woo is said to be internal on Woo. Clearly,
internal coordinates on Woo may be chosen in many different ways.
But if such a choice is made, the remaining coordinates are said to
be external with respect to Woo. For example, coordinates XI = x,
X2 = t, P2 ... 2, PI2 ... 2 (the num ber of 2's here is arbitrary)
may be chosen to be internal for an infinitely prolonged wave
equation. In fact, for this equation system (2) may be rewritten in
the form
Pull = PO'22, Va.
Another way to choose internal coordinates in this case is XI X2,
PI ... J, P21 ... 1
(the number of units in the multiindexes is arbitrary). The
restriction of a function f E g; on Woo will be denoted by f. If a
choice of
internal and external coordinates on Woo is made, then the
analytical description of 1 is as follows.
If p~ (respectively, x) is an internal coordinate, then p~ = p~
(respectively, Xj = x). If p~ is one of the external coordinates,
then p~ = <p ~( ••• p~ ... ) where <p:" is a function of the
internal coordinates. For example, if the internal coordinates on
the infinitely prolonged wave equation are chosen by the first of
the two ways described above, then
Pl1 = P22, PI12 = P222 , P111 = P122,
Pl1l1 = P2222 , etc.
SYMMETRIES AND CONSERVATION LAWS OF PDEs 7
Finally, if f = 1(···, xi"'" p~, .. . ), then f = 1(· .. , xi"'"
p~, .. . ). Operators of the form
constitute a natural class of operators on Joo, called a
C-differential. Here aCT is a s x s matrix whose elements are
smooth functions on Joo and it is supposed that d acts on columns I
= ([I , ... , Is)', Ii E gji, by the rule
d/= I aCT (I?CTh). iCTi",k DCTls
The main property of C-diflerential operators is that they are
restrictable on submanifolds of the form Woo in Joo. Namely, the
restriction i5i of Di on Woo is given by the equality
- a I' . a D·=-+ }-. I" PCTj a }, uXi i.CT PCT
where I' means that the summing is taken over all internal p~ and
it is supposed that all Xi are chosen to be internal coordinates.
Next, 15" = 15 PI 0 i5 pz 0 ••• 0 i5 P,
if (T = (PI, P2, .. " p,) and
~ = I aCT i5CT , iCTi",k
where the elements of the matrix aCT are restrictions of the
corresponding elements of a" on Woo.
The described restriction operation has an inner invariant sense
and, in particular, does not depend on the choice of internal
coordinates on Woo. Moreover, C-diflerential operators are
completely characterized by their pro perty to be restrictable on
every submanifold of the form '&" in Joo.
3. Contact Structure on Infinite Jets and Infinitesimal
Transformations
Geometrically, full derivative operators may be treated as vector
fields on Joo. Namely, the vector of the vector field corresponding
to Di and having its origin at the point (J = ( ... , xi' ... , p~,
... ) E Joo has the numbers 8ii and P~i as its components with
respect to coordinates Xi and p~, respectively. It shows that the
vectors corresponding to D J , ••• , Dn are independent at each
point (J E J=. Therefore, they generate an n-dimensional plane Co
which is tangent to J= at (J.
The field (J~ Co of n-planes on Joo is called the Cartan
distribution, or the contact structure of infinite order on J=. A
n-dimensional submanifold L c Joo is said to be integral if its
tangent plane at (J E L coincides with Co for every (J.
One can associate with every vector function 'P = ('P\(x), ... ,
'Pm(x» (m is the
8 A. M. VINOGRADOV
number of dependent variables) the n-dimensional submanifold L<p
in Joo, which is given by the following equations
L<p~l~~~~~~: p;"=--
ax" ---------
Manifold L<p is an integral one. Conversely, every integral
manifold in Joo has the local form L<p. Therefore, bearing in
mind the correspondence cp -->0 L<p, one can think that 'the
space' of all smooth vector functions cp(x) is geometrically
realized as the totality of all integral submanifolds in Joo.
Now, let <1>: Joo-->o Joo be a smooth transformation. We
will say that it conserves the contact structure on Joo if it maps
Co onto C<I>(O) for every 0 E Joo. We will call such
transformations contact ones. Evidently, every contact <I>
maps integral submanifolds into integral ones. In that case,
<I>(L<p) is an integral submanifold and, therefore, has
(locally) the form Lop for a suitable vector function 1/1. So, in
this way <I> generates a transformation cp"""" 1/1 in 'the
space' of smooth vector functions. In the following, we will need
the infinitesimal variant of the above construction.
Infinitesimal transformation Xi-->OXi + ea;,
p~-->op{,.+ea{,., where ai, a{,.E8i', may be thought of as a
vector field on JOO or, equivalently, as the operator
n iJ . iJ X= I ai-+ Ia{,.-j'
i~1 aXi j,,, iJp"
Then the condition that it conserves the contact structure on JOO
may be formulated in the form
n
[X, D;J = I Aij~, i = 1, ... , n, j~1
where [X, DJ = X 0 Di - Di 0 X and Aij are some functions on Joo.
The last equations with undetermined functions Aij may be solved
explicitly. The result is that
n
where f= (fl,'" ,fm), t, J.L; E 8i' and
a 3f = I Du(h) -a j'
u,j p"
(3)
(4)
Functions t, JLi may be arbitrary in these formulas. The operator
3f is said to be the evolution differentiation corresponding to the
generating function f.
SYMMETRIES AND CONSERVATION LAWS OF PDEs 9
Infinitesimal contact transformations of the form X = L /LiDi
transform every integral submanifold in ] 00 into itself. So, via
the above construction, they generate the identity transformation
of the space of vector functions cp(x). In that sense, these
transformations are trivial. By the reason, it is natural to
identify infinitesimal contact transformations X and Y if X - Y = I
/LiDi. In other words, we identify such contact fields which have
the common evolution part 3f in its decomposition (3).
Infinitesimal transformation in the space of vector functions cp(x)
generated by the field (3) via the above-mentioned construction
is
CPi(X) >-+ CPi(X) + Egi(X), i = 1, ... , m,
where the gi(X)'S are obtained from fi (x, u, ... , p~, ... ) via
substitutions
In other words, the field (3) generates a flow in the space of
vector functions cP (x) which is described by the following
evolution-type system:
acpi ( aluICPi(X)) - = fi x, cp(x), ... , , ... , aT axu
(5)
where T (= 'time') is a new independent variable. Commutators of
evolution differentiations are also evolution
differentiations.
More exactly,
{f, g} = ({f, gh, ... , {f, g}m).
It follows from (6) that evolution differentiations constitute a
Lie algebra with respect to the usual commutators operation as well
as its generating functions with respect to the bracket {., .}. The
latter is called the higher Jacobi bracket. The correspondence
f>-+ 3! identifies these two algebras.
4. Higher Symmetries of Differential Equations
Let <1>: Joo ~ Joo be a contact transformation. It is a
symmetry of Equations (1) if the corresponding flow on the space of
the vector functions transforms the solutions of (1) into its
solutions.
10 A. M. VINOGRADOV
The problem of finding all symmetries of a given PDE system is
equivalent to solving a new nonlinear PDE system which is, as a
rule, much more complicated than the initial one. Therefore, in
practice, one cannot use symmetry con siderations to investigate a
given equation, because it is not possible to find its symmetries
in an explicit form. However, affairs change considerably if one
adopts the infinitesimal point of view.
Namely, let us understand infinitesimal contact transformations of
Joo as vector fields of the form (3). Such a field is a higher
infinitesimal symmetry of (1) if it tangents to the infinite
prolongation Woo of (1). This is equivalent to the fact that the
flow generated by this field on the space of the vector functions
(see above) transforms solutions of (1) into themselves. Vector
fields of the form I !LjDj are tangent to every submanifold of the
form Woo in Joo. So vector field (3) is tangent to a given Woo if
and only if it does its 'evolution part' 3",. The latter is
equivalent to the fact that
(7)
(8)
Now, let us introduce the universal linearization operator IF by
putting
Taking into account the coordinate expression (4) for 3f, one can
obtain the following coordinate expression for IF:
IF = (Ii~--~~-:Iri-~~) L -1 D" ... L ---,;; D" " ap" " iip"
or, in an alternative form,
It shows that IF is a C-differential operator and therefore may be
restricted on Woo. Denote this restriction by IF. Then the equality
(8) expressing the fact that cp is generating function of an
infinitesimal symmetry of (1) may be rewritten in the form
(9)
This linear equation is the basic in the symmetry theory.
SYMMETRIES AND CONSERVATION LAWS OF PDEs 11
Let Sym uy denote the totality of all higher infinitesimal
symmetries of a PDE system uy. This is a linear space which may be
identified with the solution space of Equation (9). In the
following, we will identify infinitesimal symmetries with
generating functions or, more exactly, with their restrictions on
UYoo.
If !p, (fr are symmetries of UY, then the Jacobi bracket {cp, "'}
is also a symmetry of uy. Therefore, Sym uy is a Lie algebra with
respect to the Jacobi bracket operation. This simple fact may be
very useful within the process of solving (9) (i.e., within the
process of finding Sym UY) because it makes it possible to generate
new solutions of (9).
The commutator relation
(10)
where A is a C-differential operator on UYoo, is also very useful
in practice, as a consequence of (9). The indetermined operator A
entering into it may be evaluated in some situations. In
particular, it is equal to zero for scalar evolution
equations.
Let 91 c Sym uy be a Lie subalgebra. Solutions of (1) which don't
change under actions of infinitesimal transformations belonging to
121, are said to be l2I-invariant. If cp (x) is an \.JI-invariant
solution, then it does not change under the action of flow (5) for
every f E 21. This means that
( aiuicp ) f x,cp(x), ... ,--, ... =0. ax"
Therefore, \.JI-invariant solutions of (1) satisfy the system
F=O, h = 0, ... , fs = 0,
where {fl, ... , fs } is a basis of \'1. Evidently, the converse is
also true. This system for finding 21-invariant solutions is
overdetermined. It is much simpler to solve than the initial system
(1). Roughly speaking, under some regularity conditions, this
overdetermined system is equivalent to a PDE system with n - s
independent variables, where s is the dimension of 21.
5. Classical and Higher Symmetries
A classical symmetry of (1) is a transformation x ~ x' = f(x, u),
u~ u' = g(x, u) of dependent and independent variables which also
maps solutions of (1) into solutions. A classical infinitesimal
symmetry is a vector field on the space JO of dependent and
independent variables, say
12 A. M. VINOGRADOV
Xi ~ Xi + Eai(X, U),
maps solutions of (1) into themselves. The field Xu may be
canonically lifted up to a contact field X on Joo.
Generating function 'P = ('PI, ... , 'Pm) of X is given by the
formula
n
'Pj=bj(x,u)- L Ptak(X,U). k~1
This shows which part of 'higher' theory is the classical one.
Namely, the generating functions of higher symmetries may be
arbitrary functions of variables X, u"p~, 0<10"1<00. But the
generating functions of classical symmetries may depend only on
variables X, u, p} and, in addition, this dependence is of a
special sort.
From the classical point of view, the case m> 1 described above
differs from the case m = 1 (m is the number of dependent
variables). In the last case, infinitesimal transformation of the
space JI with local coordinates XI, ... , Xn , u, PI, ... , Pn
which conserve the 'integrability condition'
du - PI dXI - ..• - Pn dXn = 0
are to be considered. These are classical contact fields on the
space J I • Every such field has the form
X f = L --+ f- L pi- -+ L -+Pi- -, 1 af a ( n af) a n (a f af) a
i~1 api aXi i~1 api au i~1 aXi au api
where f = f(x, u, p) may be an arbitrary function on J I .
Vector field XI can be canonically lifted up to a contact vector
field X on Joo. Moreover,
X=3f+L af D i . ap;
Therefore, in the case m = 1, the classical theory is more richer
than in the case m>1.
6. Existence of Higher Symmetries
How wide is the class of equations admitting nonclassical
symmetries? At a first glance the answer is evident: this class is
much wider than the class admitting only classical symmetries. This
is because there are many more potential can didates to be a
higher symmetry for a given equation than a classical one. However,
there are no 'nondegenerated' nonlinear equations with the number
of independent variables greater than two which possess
nonclassical symmetries,
SYMMETRIES AND CONSERVATION LAWS OF PDEs 13
i.e., essentially higher symmetries. This phenomenon was first
observed during the course of studying concrete equations.
But what prevents differential equations from having symmetries? In
a more constructive form, this question means: how does one exactly
describe the 'inhomogeneosities' of diffeential equations which
make it nonsymmetric? In classical differential geometry, such
inhomogeneosities are nothing more than differential invariants
(for example, the curvature in Riemannian geometry). For this
reason, it is natural to suppose that some similar things have to
be in the same situation as those we are interested in. In fact,
the desirable analogy exists but it is necessary to use elements of
the secondary differential calculus in order to describe it (see
[16]). Of course, we have no possibility to do this here and
instead, we will give some informal explanations of why
differential equations with more than two independent variables
generally have no higher symmetries.
Let us restrict ourselves to the case of one scalar equation of the
second order. The main symbol (= the second-order part of the
left-hand side of such an equation) is the simplest of its
differential invariants. Supposing that this symbol is
nondegenerated, one can consider it as a map assigning to each of
the initial equation solution a conform metric on the space of the
independent variables. 'Conform' here means 'up to an arbitrary
functional multiplier'. But all two dimensional conform metrics
are the same. In other words, they are 'flat'. In particular, the
so-called conform curvature tensor is identically equal to zero in
that case. Contrary, in dimensions greater than two, the conform
curvature tensor is nontrivial and, moreover, other nontrivial
conform invariants exist. Then nontriviality of such invariants for
a given equation, simply reflects its nonlinearity in the
second-order derivatives. So these invariants whose non triviality
results from the nonlinearity of the symbol of the considered
equation does not allow it to be symmetric.
However, it would be a great mistake to restrict the higher
symmetries theory only to the cases of one or two independent
variables, taking into account what was said before. First of all,
the 'higher' point of view considerably simplifies the classical
theory and, in particular, simplifies calculations during the
course of finding classical symmetries. Next, a development of the
higher symmetries theory naturally leads to the discovery of
phenomena which make differential equations nonsymmetric. These are
of great importance for the theory of differential equations. And,
finally, it stimulates the search for a more general point of view
on the concept of symmetry in the field of differential equations.
In that direction, the transition on the nonlocal point of view
looks very promising. It is discussed in the paper [15] of this
volume.
7. Conservation Laws
The notion of a conservation law for a given differential equation
is a concept dual in the sense of the conception of symmetry. A
relation between them is
14 A. M. VINOGRADOV
established in some cases by the famous N6ther theorem. The long
time and very fruitful use of this theorem has lead to the
widespread opinion that every conservation law is a reflection of
some symmetry. In fact, it is not so and we will explain this
below.
The notion of conservation law is not so simple as it may appear at
a first glance. In fact, it is of a cohomological nature. In the
considered context, the latter means the following.
Let us suppose that the investigating physical substance is
described by Equations (1). A 'vector' n = (WI, ... , wn ), where
Wi = Wi(X, u, ... , p~, ... ) E ,Cfji
is said to be a conserved current for these equations and,
consequently, for the considered physical substance if
div n == L Di(WJ = 0, by virtue of (1). (11)
This is equivalent to
or to
are some C-differential operators. The standard 'physical'
interpretation of the last definition is as follows.
Suppose that one of the independent variables, say, XI, is the
'time', i.e., Xl = t. Let also au be a domain in the space of
variables X2, ••. , Xn • Then the quantity
Cn = I WI dX2 ... dXn
is a function of I. The Gaussian formula being applied to (11) (or
to (12» then shows that the difference Ca(tz) - Cn(td is equal to
the flow of the 'vector' n' = (W2, ... , wn ) through the boundary
aau of au within the time interval II:S; I:S;
t2' In particular, if that flow is trivial, then Ca(t) is a
constant, i.e., Cn(t) is a conserved quantity.
There exists a very simple way to construct conserved currents.
Namely, let aij E :!F, 1 :s; i, j:S; n, be arbitrary functions on
Joo satisfying aij = - aji and
Wi = L Dj(aij)' j
(14)
Then div n = 0 if n = (WI, ... , wn ). Therefore, n is a conserved
current for (1) as
SYMMETRIES AND CONSERVATION LAWS OF PDEs 15
well as for every other equation posed on dependent variables U\
... , urn. This shows that conserved currents of the form (14) are
nonspecific for Equations (I) and, therefore, for the physical
object described by them. In other words, they contain no
meaningful information about this object. For this reason, these
conserved currents should be noted as trivial.
Bearing all this in mind, it is natural to identify those conserved
currents which differ from each other by a trivial current. This
idea leads to the notion of conservation law. More exactly,
conserved current 0 1 and O2 are said to be equivalent if 0 1 - O2
is a trivial conserved current. An equivalent class of conserved
currents for (1) is said to be a conservation law for (1).
One must pay attention to the fact that the terms 'conserved
current' (and, sometimes, 'conserved density') and 'conservation
law' are usually identically used. But according to the previously
introduced terminology, they have quite different meanings. Also,
we will use the term 'conserved density' for the time component of
a conserved current in the case when the time coordinate is
distinguished. Below, the term 'integral of motion' is used as a
synonym to 'conservation laws'.
Not entering into the details which one can find in [4], or [17],
we mention that a class of equivalent conserved currents, i.e.,
conservation laws, is, in fact, an (n - I}-dimensional cohomology
class in the so-called horizontal de Rham com plex on Woo. This
cohomological interpretation of the conception of conservation law
is of fundamental importance because it makes it possible to apply
powerful homological methods in finding the conservation laws and
also during the course of its applications. In particular,
describing below the method of generating functions is of a
cohomological nature.
8. Generating Functions of Conservation Laws
Operator L * conjugated with a C-differential operator L = L" a"D"
is defined as
Supposing that 0 is a conserved current for (1), we define
!/In = (Ai(1), A'W), ... , A~ (1»,
where the operators Ai are taken from the equality (13). The
restriction ;jin of !/In on Woo does not change if one changes n by
an equivalent conserved current. Therefore, !/In characterizes the
conservation law containing 0 as a unity and we will call this
function the generating function of this conservation law.
The operator A entering into (13) is not uniquely defined. For this
reason, many generating functions may, in general, correspond to a
given conservation law. However, if the considered system (1) is
normal, then the following im portant fact holds: the generating
function of a given conservation law for (1)
16 A. M. VINOGRADOV
does not depend on the choice of A satisfying (13). Therefore, in
that case, the generating functions completely characterize the
corresponding conservation laws. In particular, the two
conservation laws are different from each other if and only if
their generating functions are different.
The normality condition mentioned above means that system (1) is
determined, i.e., the number of essentially different equations
entering into it is equal to the number of dependent variables, and
its main symbol is nondegenerated. In this connection, we stress
that PDE systems of practical interest are usually normal ones (see
[3,4] for more details). Therefore, the problem of finding all the
conservation laws for a given normal system is reduced to the
problem of finding the corresponding generating functions.
The main result allowing one to solve the last problem completely
in many cases is that the generating functions of the conservation
laws satisfy the following equation:
n~=o. (15)
Here It is the matrix C-differential operator conjugated with IF.
This means that It = II V' kill, where V' kl = Ll ik supposing that
IF = II Llijll (the conjugation of scalar operators was defined at
the beginning of this section). More exactly,
It = (i(~:;~;~;~~~;L{;~~-:~:-~~) u apu u apu
Comparing (9) and (15), one can see that the problems of finding
higher symmetries and conservation laws after being reformulated in
terms of generating functions, are reduced to solving conjugate
equations. This demonstrates a duality of these notions which was
mentioned earlier.
Not every solution", of (15) is, in fact, a generating function of
a conservation law for (1). In order that it would be so, the
following additional condition has to be satisfied.
An equivalent form of Equation (15) is
I}", = A(F),
where A is a matrix C-differential operator. A solution ~ of (15)
is a generating function for a conservation law of (1) if and only
if the operator IF +.4:* may be represented in the following
form
(16)
where B is a C-differential operator. So the following procedure
for finding the generating functions of conservation
laws may be devised firstly, to solve equations (15) and, secondly,
to test the obtained solutions by (16).
SYMMETRIES AND CONSERVATION LAWS OF PDEs 17
Equations (15) and (16) are, in fact, corollaries of a theorem
describing the structure of C-spectral sequences for normal
equations. The theory of C-spectral sequences allows us to
construct efficient procedures for finding the conservation laws
for equations which are not normal. However, we will not discuss
this here. Instead see [4J for more details.
It is worth paying attention to a fact which is essential for
physicists who usually describe an investigated physical object by
different systems of differential equations. Here we do not have in
mind descriptions which differ by degrees of exactitude but those
which can, in a sense, be reduced to each other. For example, such
are Eulerian and Lagrangian approaches to continuum media mechanics
as well as alternative descriptions of the electromagnetic field by
means of its stress tensor or by its potentials. The notion of
conservation law introduced above is associated with equations
describing the physical object but not with this object in itself.
For this reason, it may occur that different equations describing
the same physical situation may have different groups of
conservation laws. For example, Euler and Lagrange approaches to
the same continuum media may lead to different sets of conservation
laws. So it is not correct to speak about conservation laws of a
physical object alongside a reference to describing its equations.
This is not very satisfactory, but adopting a nonlocal point of
view can overcome this phenomenon.
Concluding, we remark that there are very interesting and important
global aspects in the conservation laws theory. See, for example,
[4J and [17].
9. Symmetries and Conservation Laws
Now we will discuss the main interrelations between symmetries and
conservation laws.
First of all, symmetries may be used to obtain new conservation
laws from some of those already known. More exactly, there exists a
natural action of symmetries on conservation laws which, in terms
of generating functions, is described as follows.
Let cp be the generating function of a symmetry of Equations (1)
and 1/1 be the generating function of its conservation law.
Because 3",( F) = () on UJJ=, 3",( F) = B( F), where B is a
C-differential opera tor. Let
Then the restriction of 3",{ I/I} on UJJ= is a generating function
of a conservation law for (1). Therefore, Iii t--'> 3", { I/I}
is an action of the Lie algebra Sym UJJ on the space of all
conservation laws of UJJ.
We now discuss the famous Nother theorems (both direct and inverse)
from the point of view of generating functions theory. The main
result in this
18 A. M. VINOGRADOV
(17)
if F = 0 is a Euler-Lagrange system, i.e., a system obtained from a
functional by means of classical variational procedure. In other
words, operator IF is self adjoint for Euler-Lagrange equations.
Comparing Equations (9) and (IS) and taking into account (17), one
can see that every generating function of a conservation law for a
Euler-Lagrange equation is, at the same time, the generating
function of its symmetry. So there exists the canonical map
{conservation laws} --------? {symmetries}
for Euler-Lagrange equations. It is not difficult to see that this
map coincides with that which is given by the inverse N6ther
theorem.
To a point, it is natural to illustrate the advantage of the
adopted approach to symmetries and conservation laws by
generalizing the inverse N6ther theorem outside the framework of
variation calculus. We will call the equation F = () conformly
selfadjoint if
(18)
where A is an operator. Evidently, every solution of Equations (15)
is also a solution of (9) for conformly self-adjoint equations.
Therefore, as above, there exists a natural map from conservation
laws into symmetries for such equations. If A 11, then the
conformly self-adjoint equation is not, as a rule, of the
Euler-Lagrange type, i.e., cannot be put into the framework of the
calculus of variations. The simplest example of this kind is the
equation Ux = u, for which A=-l.
The above consideration is not invertible, i.e., does not lead to
the map
{symmetries}------? {conservation laws}.
This is because not every solution of (15) is a generating function
of the conservation laws (conditions (16) are also to be
satisfied). However, if OY is the Euler-Lagrange equation
corresponding to a Lagrangian 5£, then every sym metry of 5£
automatically satisfies (16). It follows then from (9), (15) and
(17), that in this case the generating functions of the symmetries
of 5£ are also generating functions for the conservation laws of
the corresponding Euler Lagrange equation. In other words, we have
a natural map
Iconservation laws {symmetries of a Lagrangian}-------? of
corresponding Euler
Lagrange equations
This map is identical to the one that is given by the direct N6ther
theorem. Hamiltonian equations form another class for which a
natural correspondence
between symmetries and conservation laws exists. Not entering into
details of
SYMMETRIES AND CONSERVATION LAWS OF PDEs 19
Hamiltonian theory for partial differential equations (see [18],
where it is presen ted in the necessary general form), we recall
only the fact that a Hamiltonian structure is given by a
C-differential operator, say r. Now suppose that the equation F = 0
is Hamiltonian with respect to the Hamiltonian structure r. Then
the' operator r maps the solution of (15) into solutions of (9).
Therefore, in this case, a mapping of conservation laws into
symmetries also exists.
10. Complete Integrability Problem for Partial Differential
Equations
The notion of a conservation law in the form presented here is an
exact analog of the notion of an integral for ordinary differential
equations. It is well known that every ordinary differential
equation possesses a complete integral locally in a neighbourhood
of everyone of its regular points. Therefore, it is natural to ask
if partial differential equations also have complete integrals
locally supposing necessary regularity conditions hold? More
exactly, we are interested in such a set of conservation laws (or,
more physically, 'integrals of motion') which completely determine
the solutions of a given equation 'in small'.
Now having to hand the above-described effective tool for finding
out all conservation laws of a given PDE system, we are able to
answer this question. It is negative. This is because partial
differential equations, as a rule, have only a finite number of
conservation laws (see the subsequent papers of this volume) but
its solutions spaces are infinite-dimensional. However, the
situation does not look hopeless. The following two reasons are
sources of an optimism.
First of all, it seems to be true that normal linear differential
equations are completely integrable 'in the small' via linear
conservation laws (see [4]). Secondly, some fact forces us to
suppose that 'regular' differential equations possess infinity of
nonloeal conservation laws. By nonlocal, we mean such conservation
laws whose generating functions depend on nonloeal variables. The
exact sense of the last notion is given in [15].
11. Some Problems
]. The reader will see from the subsequent papers that calculations
which are neccssary in order to find all symmetries or conservation
laws of a given equation, are usually very cumbersome. So one of
the more important problems now is to make the existing methods
essentially more effective. Evidently, one of the ways in that
direction is to develop computer methods which will allow one to
carry out enormous routine symbolic calculations which arise at the
very begin ning in solving Equations (9) and (15). Computers were
used in some of the subsequent papers from which one may obtain
somc feelings on the subject. The state of arts in computer
applications to the above problem are discussed in the 'computer
part' of this special volume.
Another line along which important progress may be expected is
the
20 A. M. VINOGRADOV
development of differential invariants theory for differential
equations. As was mentioned, such invariants provide obstructions
to the symmetricity of differen tial equations.
Generally, further development of symmetries and conservation laws
theory will depend on progress in the geometry of differential
equations and especially in the secondary differential calculus
(see [13, 19]).
2. The most interesting question for applications is what
information about the equation under consideration may be taken
from what is known of its symmetries and conservation laws. This
problem has principle and technological aspects.
Fir:;t of all, we must state that a considerable vagueness exists
over how symmetries and conservation laws may be used in the
analysis of concrete equations. For example, the following are
waiting to be investigated in that direction: symmetry analysis of
discontinuities and singularities of solutions; the relationship
between asymptotic methods and symmetry analysis; applications of
conservation laws in stability theory and in the theory of
'intermediate states', etc.
The character of technological problems arising in the considered
domain may be illustrated by the problem of finding invariant
solutions. First of all, it is necessary to find all k-dimensional
subalgebras, k < n, in the Lie algebra Sym qy. The next step is
to obtain reduced systems of partial equations, i.e., systems with
less than n number of independent variables whose solutions are
invariant solutions of the initial equations. Finally, these
reduced systems are to be solved. It is clear that all this can be
carried out in detail only with aid of the appropriate computer
machinery (see [7] for an illustration).
3. It is natural to suppose that both symmetries and conservation
laws theories may be built for more general equations than
differential ones. For example, integro-differential equations are
surely such ones. One can imagine that sym metry transformations
for such equations will be operators which are more general than
differential. If this is the fact, more general symmetry trans
formations than described before exist for differential equations.
This is because the latter may be understood, for example, as
integro-differential ones. This leads us to assume the existence of
more general frames for symmetry theory.
These considerations, as well as some others, force us to
generalize the local theory presented above. Essentially, this
problem is to find an exact definition for the conceptions of
'nonlocal' symmetry and conservation laws. An approach to it is
made in the paper [15] of this volume. It bases on the notion of
covering in the category of differential equations. We point out
that only the first steps are made in that direction and the
appearing theory looks attractive because of its nonstandard
features.
References
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nonlinear differential equation and algebra-geometric foundations
of Lagrangian field theory with constraints, Dokl. Acad. Nauk. SSSR
238 (1978), 1028-1031 (English trapslation in Soviet Math. Dokl. 19
(1978), 144-148).
SYMMETRIES AND CONSERVATION LAWS OF PDEs 21
2. Vinogradov, A. M.: The theory of higher infinitesimal symmetries
of nonlinear partial differential equations, Dokl. Acad. Nauk. SSSR
248 (1979), 274-278 (English translation in Soviet Math. Dok!. 20
(1979), 985-990).
3. Vinogradov, A. M. Local symmetries and conservation laws, Acta
Appl. Math. 2 (1984), 21-78. 4. Vinogradov, A. M.: The C-spectral
sequence, Lagrangian formalism and conservation laws, J.
Math. Anal. Appl. 100 (1984), 2-129. 5. Ibragimov, N. H.:
Transformation Groups Applied to Mathematical Physics, D. Reidel,
Dor
drecht, 1985. 6. Oliver, P. J.: Application of Lie Groups to
Differential Equations, Springer, New York, 1986. 7. Gusyatnikova,
V. N., Samokhin, A. V., Titov, V. S., Vinogradov, A. M., and
Yumaguzhin, V.
A.: Symmetries and conservation laws Kadomtsev-Pogutse equations,
Acta App/. Math. 15 (1989), 23-64.
8. Gusyatnikova, V. N. and Yumaguzhin, V. A.: Symmetries and
conservation laws of Navier Stokes equations, Acta App!. Math. 15
(1989), 65-81.
9. Titov, V. S.: On symmetries and conservation laws of the
equations of shallow water with an axisymmetric profile of bottom,
Acta App/. Math. 15 (1989),137-147.
10. Verbovetsky, A. M.: Local nonintegrability of the long-short
wave interaction equations, Acta Appl. Math. 15 (1989),
121-136.
11. Sharomet, N. 0.: Symmetries, invariant solutions and
conservation laws of the nonlinear acoustics equation, Acta App/.
Math. 15 (1989), 83-120.
12. Romanovsky, Yu. R.: On symmetries of the heat equation, Acta
App/. Math. 15 (1989), 149-160.
13. Krasil'shchik, I. S" Lychagin, V. V., and Vinogradov, A. M.:
Geometry of Jet Spaces and Nonlinear Partial Differential
Equations, Gordon and Breach, New York, 1986.
14. Senashov, S. I. and Vinogradov, A. M.: Symmetries and
conservation laws of 2-dimensional ideal plasticity. Proc.
Edinburgh Math. Soc. 31 (1988), 415-439.
15. Krasil'shchik, I. S. and Vinogradov, A. M.: Nonlocal trends in
the geometry of differential equations: symmetries, conservation
laws, and Backlund transformations, Acta App/. Math. 15
(1989),161-209.
16. Gusyatnikova, V. N., Vinogradov, A. M., and Yumaguzhin, W. A.:
Secondary differential operators, J. Geom. Phys. 2 (1985),
23-66.
17. Tsujishita, T.: On variation biocomplexes associated to
differential equations, Osaka Math. J. 19 (1982), 311-363.
18. Astashov, A. M. and Vinogradov, A. M.: On the structure of
Hamiltonian operators in field theory, J. Geom. Phys. 3 (1986),
264-2~n.
19. Vinogradov, A. M.: The category of differential equations and
its significance for physics, Proc. Conf. Differential Geometry and
its Applications, Nove Mesto na Morave (VSSR), 5-9 September 1983,
Part 2, pp. 289-301.
Acta Applicandae Mathematicae 15: 23-64, 1989. © 1989 Kluwer
Academic Publishers.
Symmetries and Conservation Laws of Kadomtsev-Pogutse
Equations
(Their computation and first applications)
V. N. GUSYATNIKOVA
Program Systems Institute of the U.S.S.R. Academy of Sciences,
152140, Pereslavl-Zalessky, U.S.S.R.
A. V. SAMOKHIN Moscow Institute of Civil Aviation Engineers,
Moscow, U.S.S.R.
V. S. TITOV Program Systems Institute of the U.S.S.R. Academy of
Sciences, 152140, Pereslavl-Zalessky, U.S.S.R.
A. M. VINOGRADOV Moscow State University, Department of Mechanics
and Mathematics, 117234, Moscow, U.S.S. R.
and
V. A. YUMAGUZHIN Program Systems Institute of the U.S.S.R. Academy
of Sciences, 152140, Pereslavl-Zalessky, U.S.S.R.
(Received: 11 April 1988)
Abstract. Kadomtsev-Pogutse equations are of great interest from
the viewpoint of the theory of symmetries and conservation laws
and, in particular, enable us to demonstrate their potentials 'in
action'. This paper presents, firstly, the results of computations
of symmetries and conservation laws for these equations and the
methods of obtaining these results. Apparently, all the local
symmetries and conservation laws admitted by the considered
equations are exhausted by those enumerated in this paper.
Secondly, we point out some reductions of Kadomtsev-Pogutse
equations to more simpler forms which have less independent
variables and which, in some cases, allow us to construct exact
solutions. Finally, the technique of solution deformation by
symmetries and their physical inter pretation are
demonstrated.
AMS subject classifications (1980). 35Q20, 58F07, 35G20.
Key words. Magnetohydrodynamic equations, symmetry, conservation
laws, invariant solution.
o. Introduction
In 1973, B. B. Kadomtsev and O. P. Pogutse [1] suggested the now
well-known simplification of the general system or
magnetohydrodynamic (MHD) equations, discarding some details,
unimportant from the viewpoint of the problem of holding a high
temperature plasma in 'tokamak'-type facilities. They proceeded
from the ideal MHD equations, since the characteristic times of
most important
24 V. N. GUSYATNIKOVA ET AL.
physical processes are much less than the distinctive time of
dissipation stipulated by plasma electric resistance and viscosity.
Besides, it had been taken into account that for the plasma
stability
(1) it is necessary that plasma pressure and the cross-component of
magnetic field pressure are both much less than the pressure
created by the longitudal component of a magnetic field;
(2) it is expedient that the lesser radius of a tokamak torus
should be much less than the greater one. As a result, the system
of the two scolar equations was obtained. After proper norming,
they have the form
(1)
Here the functions cp and '" are the potentials of the velocity and
the cross-component of the magnetic field (they may be also
interpreted as the potential of the electric field and of the
vector potential z-component of the magnetic field). The coordinate
system (x, y, z) is such that the z-axis coincides with the
'tokamak' axis. Besides,
and [u, v Jz = uxvy - uyvx is a z-component of the vector product
[u, v J of vectors u, v.
System (1) will be referred to as the Kadomtsev-Pogutse equations.
This system has become popular in the West since the publications
of White et at. (1974) [2J and Strauss (1976) [3J and often figures
under the name 'reduced MHD equations' or 'Strauss
equations'.
Simplicity and cleanness of Kadomtsev-Pogutse equations (in
comparison with the full MHD equations) have made it possible to
construct, on their base, a theory of the kink and tearing
instabilities (see [4J), and a theory of the first mode
reconnection which has been given experimental support. Their usage
has also made it rather easier to investigate the instability by
approximate methods [2].
Kadomtsev-Pogutse equations are of great interest from the
viewpoint of the theory of symmetries and conservation laws and, in
particular, they enable us to demonstrate their potentials 'in
action'. This paper presents the results of computations of
symmetries and conservation laws for Equations (1) and the methods
of obtaining these results. Apparently, all the local symmetries
and conservation laws admitted by Equations (1) are exhausted by
those enumerated in this paper. We also point out some reductions
of the Kadomtsev-Pogutse equations to more simpler forms than those
which have less independent vari-
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 25
abIes and which, in some cases, allow us to construct exact
solutions. Finally, the technique of solution deformation by
symmetries and its physical interpretation are demonstrated.
Investigation of Equations (1) from this point of view was
initiated by A. V. Samokhin [6].
This should point out the Marsden-Morrison work [7], where it was
shown that the system (1) may be interpreted as a Hamiltonian one
in some weakened sense. It may become useful in future
investigations of the Kadomtsev-Pogutse equa tions by methods of
geometric theory of differential equations in the spirit of the
book [8].
1. Symmetries
In this section, all the symmetries of Kadomtsev-Pogutse equations
whose generating functions depend on derivatives up to the first
order are described. A number of considerations indicate that
hardly any symmetries were obtained which depended on derivatives
of a higher order.
Even in the stated assumptions, a search for symmetries is
connected with calculations of great volume, though of a routine
character. In this connection, we have used the computer system
'SCoLAr' (see [9]) and, as a result, it is shown that generating
functions of the symmetries of Kadomtsev-Pogutse equations linearly
depend on the first-order derivatives. (We do not dwell upon our
interaction with the computer. This will be explained in some
detail in the last section). Remember, that the computations of
symmetries in the paper [6] have been carried out assuming the
linear dependence of the generating function on the first-order
derivatives. A description of this computation is presented below.
We emphasize that all symmetries of Kadomtsev-Pogutse equations
obtained this way prove to be classical.
When computing symmetries of low orders, it is more convenient to
use the following notations for coordinate functions on jet
manifolds. Firstly, let us denote Xl = X, Xz = y, X3 = Z, X4 = t,
Ul = 'P, Uz = 1/1. Secondly, instead of the symbols P ~, P~, we
shall write 'P x'yizk,l and 1/1 x'yizk,l correspondingly where (I =
xiyizkt', I(II = i + j + k + I. By these notations,
Kadomtsev-Pogutse equa tions come to a form C&.
(1.1)
Fz = 'P x2, + 'P y2, + 'Px'P x2y + 'Px'P y3 - 'Py'P x3 - 'Py'P xy2
-
- I/Ix2z - I/I y2z - I/Ixl/lXy2 - I/Ixl/ly3 + I/Iyl/lx3 +
I/Iyl/lxy2 = o. (1.2)
As the external coordinates on infinitely prolonged equation
C&OO, we take variables of the form I/IxiyiZk,1 for I;;. 1 and
'Pxiyizk,. for j;;. "2, I;;. 1. Indeed, the equations Fl = 0 may be
written in the form
26 V. N. GUSYATNIKOVA ET AL.
0/, = C{!z - C{!xo/y + C{!yo/x .
Noting that o/,u = D( 0/,) = C{!zu + Du( - C{!xo/y + C{!yo/x), by
induction in the order of entry of variable t into multi-index a,
one can prove that variables o/,u on Woo are expressed in terms of
o/p, C{!T' P not containing t. Similarly, by rewriting the equation
F2 = 0 to the form
C{! yZ, = - C{! xZ, - C{!xC{! xZy - C{!xC{! y' + C{!yC{! x3 +
C{!yC{! xyZ + + o/x2 z + 0/ y2z + o/xo/ xZy + o/xo/y3 - o/yo/x3 -
o/yo/ xyZ,
one can verify that functions C{! xiyizk", j ~ 2, I ~ 1 may be
expressed in terms of the internal coordinates on Woo which are
functions x, y, z, t, C{!, 0/, C{!T> o/u, where a = xiyi Zk for
any i, j, k ~ 0 and T = xiyi Zktl for any i, k ~ 0 and 0,,; j,,; 1
+(1/21).
Since the Kadomtsev-Pogutse system contains two dependent
variables, C{! and 0/, the generating function of its symmetries
are two-component. We denote it <I> = (;::). In view of what
is stated above, here we seek the generating function of a special
kind
8' = S + AC{!x + BC{!y + CC{!z + EC{!, .
:Y = T + Ao/x + Bo/y + Co/z + Eo/,.
Here A, B, C, E, T, S are functions on ,0, i.e., functions of
variables x, y, z, t, C{!, 0/. Passing to the internal coordinates,
we get
g = S + AC{!x + BC{!y + CC{!z + EC{!, .
;!j = T + Ao/x + Bo/y + Co/z + E( c{!z - C{!xo/y + C{!yo/x).
For Kadomtsev-Pogutse equations, the operator of universal
linearization IF, F = (~~) has the form
I - ( - Dz + o/yDX - o/xDy D, + C{!xDy - C{!yDx ) F - D,(D; + D;) +
R(C{!) - Dz(D; + D;) - R(o/)
and its restriction on Woo is
r = ( - Dz + o/yDx - o/xDy D, + C{!xDy - C{!yDX ) F D,(D~ + D;) +
R(C{!) - Dz(D~ + D;) - R(o/) .
In these formulas
for (= C{! or 0/. The operators Dx, Dy, D., D" in their turn, are
given by the following formulas
- iJ L iJ L- iJ --+ - -+ -Dx - .> C{!XT .> o/xu .>.1. ' uX
T U'P'T U U'¥U
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 27
- 0 I 0 I- 0 D --+ - -+ -y - " 'PYT " I/Iw 0,1, ' uy T U'PT U
'i'u
- a I 0 I 0 D --+ - -+ - z - " 'PZT " I/Izu 0,1, '
uZ 'T u'P" (T 'Yu
- a I 0 I- 0 D --+ - -+ -, - " 'P'T 0 I/I,u 0,1, ' ut 1'" 'P'T (T
'Yu
where T and u are multi-indexes of the chosen internal coordinates
on C[/Joo.
Since [pel> = (/p<l» an equation [pel> = 0 for the
generating functions of sym metries in that case, may be written
in the form
(-Dz + l/IyDX -l/IxDy)[f+(D, + 'PxDy - 'PyDx)fJ= 0 on C[/Joo,
(1.3)
(D,(D~+D;)+R('P)W+(-DAD~+D;)-R(I/I))fJ=O on C[/Joo. (1.4)
As [f and fJ depend on the first-order derivatives, the left-hand
side of Equation (1.3) linearly depends on the variables 'Pu, I/Iu,
lui = 2, while the left-hand side of 0.4) linearly depends on 'Pu,
I/Iu, lui = 4. This follows im mediately from the description of
the total derivation operators Dx, Dy, Dz , D,. For example,
~[f=~S+~~A+~~B+~~C+
+ 'P,DzE + A'Pxz + B'Pyz + C'Pzz + E'P,z . (1.5)
After restricting the left-hand sides of Equations (1.3) and (1.4)
on C[/Joo,
variables 'Pu, I/Iu correspondingly vanish for lui = 2 and lui = 4.
This follows from the next simple consideration. Variables 'Pu,
I/Iu, lui = 2 are the highest-order derivatives entering the
left-hand side of (1.3) and 'Pu, I/Iu, lui = 4 are the
highest-order derivatives in (1.4). But the highest-order
derivatives in the expression
may only occur in summands of the form
which coincide with
That coincidence follows from the general assertion
[ '" aF'; .] = L... --j P!rx, = DsF. <T,j ap <T
But D<TF; = 0, i = 1,2, on 6JJ=. It implies that summands (1.6)
turn into zero on 6JJ=, i.e., that the highest derivatives in (1.3)
and (1.4) vanish after restriction on 6JJ=. We also stress that the
same argument shows that functions A, B, C, E in (1.3) and (1.4)
vanish on 6JJ= and only their derivatives remain.
Thus, Equation (1.3), after restricting on 6JJ=, contains only the
first-order derivatives, entering polynomially. A power of
homogeneous monoms varies from zero to four. Indeed, the total
derivatives of functions S, T, A, B, C, Eon '1P linearly depend on
variables 'P<T, o/<T, lal = 1, as follows by the structure of
the total derivation operators. Therefore, Equation (1.3), after
restricting on 6JJ=, contains monoms in variables i{!<T,
,fr<T, lal = 1, of a power of no more than three. For instance,
the second summand in (1.3), o/yDxY' contains monom o/y'Pxo/xC",.
While turning to the internal coordinates, the homogeneous power
may rise according to the relation ,fr, = cpz - 'Pxo/y + 'Pyo/x .
It follows from computation that no monoms of a power more than
four arise and after reduction of similar terms, only monoms of the
power 0, 1,2,3 remain. There are 42 such monoms and their
coefficients must be identically zero. This brings to us 14
independent conditions on the functions S, T, A, B, C, E
Tz = Sz;
T",=E,;
- Sx = B,; C, = S'" ; Cy = A", ;
A", = E", ; C'" = E", ; B", = - Cx ; B", = - Ex;
E, + T", = Cz + S"'; Ax + T", = - By + Cz .
(1.7)
Further on, the variables 'P<T, r/Io., 1 ~ lal ~ 3 enter into
Equation 0.4} poly nomially, and for lal = 3, enter linearly.
Indeed, summands of the form (1.6) vanish on 6JJ=, as it was
pointed above, so the variables 'P<T, o/<T' I al = 3 enter
only the terms D<TS, D<TT, lal = 3;
and
DpK' D<T'PT' DpK' D<To/T' for Ipl = ITI = 1, lal = 2, K=A, B,
C, E;
[( rp x2y + rp y3 }Dx - (rp xy2 + rp x3 }Dy]Y',
[(o/x2y + o/y3}Dx - (o/x 3 + o/xy2)Dy];J.
(1.8)
Variables rp<T, o/<T, lal = 3 enter all the enumerated terms
linearly. Since the
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 29
third-order derivatives in the equation F2 = 0 are also linearly
related, then, when turning to the internal coordinates in (1.4),
we have the linearity in these variables being preserved.
Now the coefficients by 'P,,, I/Iu, lui = 3 in (1.4) must be
identically zero. We point out that these coefficients themselves
are polynomials of 'Pu, I/Iu, 10'1 = 1, as it follows from the
expressions (1.8). For instance, the coefficient by 'Pxy2 has the
form (after reduction of similar terms)
This polynomial of the first derivatives must be identically zero
which, in turn, means that all the coefficients by 'Pu, I/Iu, 10'1
= 1 and the free terms must be identically zero.
Thus, only the consideration of variables 'Pu, I/Iu, lui = 3 in
Equation (1.4) results in a very large number of conditions.
Discovering these is the most toilful part of the whole work of
symmetry computation. By adding the conditions obtained in this way
the relations (1.7), and after some simplification, we compile the
following list
A",=A",=O;
Sy=A,;
T",=S",;
(1.9)
System (1.9) is strongly overdetermined. Thanks to that, it is not
hard to get the general solution
S = (a' + {3'- C1)'P + (a' - (3')I/I + !F,(x2 + y2) - xH, + yO, +
R,;
T = (a' - {3')'P + (a' - (3' + C1)I/I +!FAx2 + l) - xHz + yOz +
Rz;
A = !C2x + yF+ 0; B = !C2y - xF+ H; (1.10)
C = (a + (3) + ( C2 - C1) t + C3 ; E = (a - (3) + ( C2 - C1) t + C4
;
Here a = a(z + t), {3 = (3(z - t), H(z, t), F(z, t) and R(z, t),
O(z, t) are arbitrary functions, Ci - arbitrary constants,
and
'(Y) = da{C} a ~ de'
Sufficiently rigid necessary conditions (1.10) make it possible to
avoid further monotonous treatment of polynomials in the variables
'Pu, I/Iu, 10'1 < 3. All the remaining calculations reduce to
the substitution of (1.10) into (1.3), 0.4). This results in only
one additional condition
F.,= Fzz .
30 V. N. GUSYATNIKOVA ET AL.
Whem;e, it follows F(z, t) = y(z + t) + 8(z - t), with y and 8
being arbitrary functions. Thus, the following result is
stated
THEOREM. All the symmetries of Kadomtsev-Pogutse equations whose
generat ing functions depend on the derivatives of the order ~ 1
are classical. The algebra Sym 0Jj of the classical symmetries is
generated as a linear space over IR by symmetries with generating
functions of the form
(1) s'l = (a'(cp + 1/1) + a(cpz + CPt)) " a'(cp + 1/1) + a(l/Iz +
I/It) ,
(1.11)
(2) 00 = (/3'( cP - 1/1) + /3( cpz - CPt)) f3
/3'(cp-I/I)+/3(l/Iz-l/It) '
(1.12)
(3) ~ = C'(x2 + y2) + 2y(ycpx - xcpy)) "Y y'(x2 + y2) + 2 Y(YI/Ix -
xl/ly) ,
(1.13)
(4) g; _ C 8'(x2 + l) + 28(ycpx - xcpy)) 8 - 8'(x2 + y2) + 28(yl/lx
- xl/ly) ,
(1.14)
(1.17)
(1.18)
(10) 2= (CP + Z'Pz + t'Pt) 1/1+ zl/lz + tl/lt '
(1.20)
(11) At = CCPx + Y'Py + 2z'Pz + 2t'Pt) xl/lx + yl/ly + 2zl/lz +
2tl/l, '
(1.21)
Here a = a(z + t), /3 = /3(z - t), y = y(z + t), 8 = 8(z - t), G =
G(z, t), H =
H(z, t), K = K(z, t) are arbitrary functions and
'W=daW a d( , /3'( '11) = d~~ '11) , 'w = dy«()
Y d( ,
Gt = aG(z, t)
'11 d'11' az ,
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 31
Now we describe the symmetries themselves, that is, the vector
fields on J O
which correspond to the above listed generating functions.
Let
a a a a a a X = al-+ a2 -+ a3-+ a4 -+ {31-+ {32-'
ax ay az at acp at/!
(The functions aj, (3j depend on x, y, z, t, cp, t/!). The
correspondence in question is realized by the formulas f! = X ~ UI
, ff = X ~ U2 , where
UI = dcp - CPx dx - cpy dy - cpz dz - cp, dt, U2 = dt/!- t/!x dx -
l/Iy dy - t/!z dz - t/!, dt
is a system of differential forms defining the Cartan distribution
on J O• We have
f!= {31 - alCPx - a2CPy - a3CPz - a4CP" ff = {32 - a1 t/!x - a2t/!y
- a3t/!z - a4t/!"
so that {31 = S, (32 = T, al = -A, a2 = -B, a3 = -C, a4 = -E. Let
us denote by X,,> a vector field corresponding to the generating
function <1>. Then it follows that
(1) xcA.=a'(cp-t/!)C: -aat/!)-a(aaz +~),
(2) X(;g~ = (3'(cp - t/!) (a: - a:) - a (a: -~),
(3) XCf,;, = y'(x2 + y) C: + a:) - 2y (y a: - x aay) ,
(4) Xc", =8'(X2+l)(~-~)-28(Y~-X~), • at/! acp ax ay
(5) x -~ '!J> - az'
'fio 'acp z at/! ax'
a a a (8) X= =-xH--xH --H
~H , acp Z at/! ay'
a a (9) X= =K -+K -
.flK 'acp zat/!'
Xx = cp-+ t/!-- z-- t-, acp at/! az at
<. ;>
a a a a (11) X.A,t=-x--y--2z--2t-.
ax ay az at
In conclusion of this section, we give in Table I the commutators
for the above-listed generators of the algebra Sym. (i.e., we write
down their pairwise Jacobi brackets, see [8]).
2. Three-Dimensional Subalgebras of Classical Symmetries
Algebra
As is known from the general theory (see [10]) the task of finding
the solutions of a system qy of differential equations which are
invariant with respect to a five-dimensional symmetry subalgebra of
qy, reduces to solving a system with n-s independent variables (n
being the number of independent variables in qy).
If, following this remark, we want to select those solutions of the
Kadomtsev Pogutse equations whose findings are reduced to
integrating ordinary differential equations, then we have to
discover all three-dimensional subalgebras in the symmetries
algebra of these equations.
A description of all three-dimensional symmetries subalgebras of
the Kadom tsev-Pogutse equations is a rather cumbersome task. It
compels us to restrict this task in the following way. Namely, we
take a symmetries subalgebra d for the Kadomtsev-Pogutse equations
which is generated by d"" OOtl, CfJ'Y' qj;s, 'l:, :¥, CfiG , 'lJeH
, X K , .'£, At while choosing all functional parameters a, (3, y,
8, G, H, K to be identically unit. Thus,
d=(C, E, F, G, H, L, M),
where
L= . ( u + Zllx + tu,) v + zVx + tv, '
M= (-2n + XUx + YUy). -2v + xVx + yUy
The symbol (P, ... , Q) is used here to denote the Lie algebra
generated by P, ... , Q. We shall describe every three-dimensional
subalgebra .'£ of the algebra d. This task divides itself naturally
in~o four cases according to the dimension of the intersection of
the required subalgebra .'£ with the commutant d 1 of the algebra
d. We recall that the commutant d 1 of the Lie algebra d is its
ideal generated by all elements of the form [a, b], a, bEd.
34 V. N. GUSYATNIKOVA ET AL.
Using the table of commutators for the generators of d
C E F 0 H L M
C 0 0 0 -2H 20 0 0 E 0 0 0 0 0 -E 0 F 0 0 0 0 0 -F 0 0 2H 0 0 0 0 0
-0 H -20 0 0 0 0 0 -H L 0 E F 0 0 0 0 M 0 0 0 0 H 0 0
one can deduce that its commutant d 1 is generated by E, F, 0, H,
i.e.,
d 1 = (E, F, 0, H),
besides, d l is Abelian.
Case 1: dim:;en d l = 3 ~:;ec d l
Since commutant d l in this case is Abelian, it follows that any
three-dimensional subspace of d is also a subalgebra. Hence, any
three-dimensional subalgebra is generated by three generators
for
ei, Ii , qi, hi E IR, i = 1,2, 3,
They have to satisfy the only condition
It is possible to reduce this matrix by elementary line
transformations to one of the following four forms
[~ 0 0
h'] [~ 0 gl
H 1 0 h2 ; 1 g2 0 1 h3 0 0
[~ II 0
0 0 0 0
Accordingly, the set of all three-dimensional subalgebras divides
itself into four nonintersecting classes
2:'1 = (E + hlH; F+ h2H; G+ h3H), hI, h2, h3 E IR;
2:'~ = (E + gl G; F + g2 G; H), gl, g2 E IR;
2:'~ = (E + /IF; G; H), II E IR;
2:'1 = (F; G; H).
Case 2: dim 2:' n .sill = 2 In this case the following generators
of a subspace 2:' may be chosen
VI = eC+ IL+ mM+ elE+ IIF+ glG+ hlH;
V2 = e2E+ hF+ g2G+ h2H;
V3 = e3E+ hF+ g3G+ h3 H ,
where e, I, m, ei, J;, gi, hi E IR, i = 1,2,3, vector (e, I, m) is
nonzero, and
rank (e2 h g2 h2) = 2. e3 h g3 h3
(2.1)
Since {Vi, Vj} E .sill and .sill is Abelian ideal, it follows that
VI, V2, V3 generate a subalgebra if and only if
{VI,V2}=r'V2+p'V3, r,pEIR,
{VI, V3} = q . V2 + s . V3, q, S E IR,
Taking (2.1) into account, we get the following system of
equations
le2 = re2 + pe3 ;
Ih= rh+ ph;
le3 = qez + se3 ;
Ih = qh+ sh;
By elementary line transformations, the matrix
( e2 h g2 h2) e3 h g3 h3
may be reduced to one of next six forms
( 00 1 0 o 1
(1 h g2 0). o 0 0 1 '
( 0 0 1 0) o 0 0 1 .
(2.2)
Accordingly, the solving of system (2.2) divides into six subcases.
We shall
36 V. N. GUSYATNIKOVA ET AL.
consider in detail the first one (that is, for e2 = 13 = 1). (All
the other subcases are studied quite analogously and, for that
reason, we do not consider them and give the final result right
away: see the list of all three-dimensional subalgebras of .st1.,
below. We note here that there are no subalgebras in the fourth
subcase.)
In the first subcase, one may obviously suppose that e1 = It = O.
Then system (2.1) takes the form
1= r; 0 = q;
0= p; 1= s;
or
This system implies that if the determinant
Im-I 2c I 2c m-I
= (m-l)2+4c2 =f 0,
then g2 = g3 = h2 = h3 = O. But this determinant is nonzero either
if c =f 0 or if c = 0 and I =f m. Accordingly, we get subalgebras
of the following two types
21.1 = (C+ IL+ mM+ glG+ hIH; E; F), I, m, gl, hI E R;
21.2 = (IL+ mM+ gIG+ hlH; E; F), I =f m, gl, hi E R.
If
1m -I 2c 1=0 2c m -I '
then c = 0, m = 1=0 and we get a subalgebra of the type
21.3 = (L+ M+ glG+ hlH; E+ gzG+ h2H; F+ g3G+ h3H), g;, II; E R, j =
1,2,3.
Case 3: dim 2n.st1.1 = 1 In this case, one may choose as the
generators of a subspace I£ the following symmetries
VI = clC+ IIL+ mIM+eIE+ I1F+ glG+ hlH;
Vz = czC+ IzL+ m2M + ezE+ hF+ g2G+ h2H;
V3 = e3E+ hF+ g3G+ h3H,
where
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 37
and vector (e3, h, g3, h3) is nonzero. Since {Vi, Vj} E d l and .:£
n d l = (V3), it follows that elements VI, VZ, V3
generate a subalgebra if and only if
{VI,vZ}=r·v3, rER,
{vz, V3} = q . V3, q E R.
Matrix
(2.3)
may be reduced by elementary line transformations to one of the
following three forms
(1 II 0). o 0 1 '
(0 1 0) o 0 1 .
Vector (e3, h, g3, h3) may be chosen in one of the following four
forms
Accordingly, solving (2.3) divides into 12 subcases. Each of them
is studied by analogy with subcases of the case .:£ n d l = 2. So
we formulate the final result right away (see the 'list of
three-dimensional sub algebras of d', below).
Case 4: dim.:£ n d l = 0 In this case, one may choose as the
generators of subspace .:£, the following symmetries
VI = C+ elE+ [IF+ gl G+ hlH;
Vz = L + ezE + fzF + gz G + hzH;
V3 = M+ e3E+ hF+ g3G+ h3H,
where el, [I, gl, hi E R, i = 1,2,3. Since {Vi, Vj} E d l and .:£ n
d l = (0) it follows that element VI, VZ, V3 generate a subalgebra
if and only if
Solving this system, we get subalgebras on a unique type
.:£0 = (C+ glG+ hlH; L+ ezE+ fzF; M-!hIG+!gIH), ez, fz, gl , hi E
R.
38 V. N. GUSYATNIKOVAET AL.
LIST OF ALL THREE-DIMENSIONAL SUBALGEBRAS OF ALGEBRA JIl
dim.:e n Jill = 3
.:e~ = (E + hlH; F+ hzH; G+ h3 H), h; E R, i = 1, 2,3;
.:e~ = (E+ giG; F+ gzG; H), g; ER, i = 1,2;
.:e~ = (E + /JF; G; H), II E R,
.:e~ = (F; G; H).
dim.:e n Jill = 2
.:ei.l = (C+ lL+ mM + gl G+ hlH; E; F), " m, gl, hi E R;
.:ei.z = (lL+ mM + gl G+ hlH; E; F), l = m, gl, hi;
.:ei.3 = (L + M+ gl G+ hlH; E + g2G+ hzH; F+ g3G+ h3 H), g;,h;ER,
i=1,2,3;
.:e~.1 = (L+ M+ IIF+ hlH; E+ fzF+ hzH; G± H), \
fl,f2,h1,h2ER;
.:eL = (L+ mM+ IIF+ hlH; E+ fzF; G± H), m =f 1,/I,fz, hi ER;
.:e~.3 = (C+(m'F2)L+ mM+ IIF+ hlH; E+ fzF+ h2H; G± H), m,fl'fz, hi,
hzER;
.:e~.4 = (C+ lL+ mM + /JF+ hlH; E+ fzF; G± H), m -l =f ±2,
" m,/J,fz, hi ER;
.:e~.l = (L + M + /JF+ gl G; E+ fzF+ gzG; H), II'fz, gl,
gzER;
.:e~.z = (lL+ mM+ /JF+ giG; E+ fzF; H), l =f m, " m, /1, Iz, gl E
R;
.:e~.l = (L+ M + elE + gl G; F+ gzG; H), el, gl, gz E R;
.:e~.z = (lL + mM + elE + gl G; F; H), l =f m, " m, el, gl E
R;
.:e~.1 = (cC + lL + mM + elE + /IF; G; H), c, " m, el, /1 E
R.
dim.:e n Jilt = 1
t ( mZ(mlgl - 2hl) .:e1.1 = C+ mlM+ glG+ hlH; L+ mzM+ fzF+ z
G+
ml +4
ml, mz, fz, /3, gl, hi E R;
mz(2g1 + mlhl) H ) + mi+4 ;F ,
ml, m2, e2, gl, hi E R;
LAWS OF KADOMTSEV-POGUTSE EQUATIONS
2?L.I = (C+ IIL+ IIF+2hzG-2gzH; M+ gzG+ hzH; E+ /JF),
11'/1,/J,gz,hz ER,1110;
2?L.z = (c+ IIF+2hzG-2gzH; M+ fzF+ gzG+ hzH, E+ /JF),
/I,fz,/J,gz,hzER, fz10;
2?tl.3 = (c+ IIF+2h2 G-2g2H; M+ g2G+ h2H; E+ /JF), II,/J, g2,
hzER;
2?i.2.1 = (C+ IlL + e1E + 2hzG- 2g2H; M + g2G+ h2H; F), II, el ,
g2, h2 E R, II 1 0;
2?i.2.2 = (C+ elE +2h2G- 2gzH; M + ezE + g2G+ hzH; F), el, ez, gz,
hz E R, ez1 0;
2?i.Z.3 = (C+ elE+ 2hzG- 2g2H; M + gzG+ hzH; F), el, g2, hz
ER;
2?b = (L + IIF; M + gzG+ hzH; E + /JF), 11,/J,g2,hz ER;
2?t2 = (L + elE; M + gzG+ hzH; F), el, gz, h2 E R;
2?t3 = (L+ elE+ /IF; M+ hzH; G+ h3H), el, /I, hz, h3 E R;
2?j.4 = (L + elE + IIF; M + gzG; H) el, II, gz E IR.
dim 2? n s1.1 = 0
2?0 = (C+ glG+ hlH; L+ e2E+ fzF; M-1hIG+!gIH), ez, fz, gz, h1 E
IR.
3. Solution Invariant with Respect to Three-Dimensional
Subalgebras
39
In this section, we describe a system of ordinary differential
equations which are to be integrated to obtain invariant solutions
for the three-dimensional subalge bras of Section 2. A consistent
and the systematical studying of all these subalgebras may be
substantially simplified, due to a particular consideration. For
this simplification, we divide all subalgebras into four classes
which are in vestigated separately.
Class 1 Here we use the following observation. It is evident that
any solution invariant with respect to the subalgebra 2?
(2?-invariant), are invariant with respect to any symmetry
belonging to 2? Therefore, whenever it is possible to obtain
explicit solutions invariant with respect to some symmetry S, the
task of finding the solutions, invariant with respect to the
algebras containing 5£ greatly simplifies.
The symmetries S = aC + hH + gG, where a, h, g are arbitrary
constants, happen to be the symmetries of such a kind for the
Kadomtsev-Pogutse equa tions. The three-dimensional subalgebras of
Section 2 containing S are
40 V. N. GUSYATNIKOVA ET AL.
Let us describe the procedure for finding S-invariant solutions of
the Kadomtsev-Pogutse equations in more detail. To obtain them, as
follows from the general theory, one has to supplement system (1.1)
with the equations
2a(ycpx - xCpy) + gcpx + hcpy = 0,
2a(y"'x - X"'y) + g"'x + h",y = ° and to solve the resulting system
of the four equations.
Rewriting system (3.1) to a form
(2ay + g)cpx + (-2ax + h)cpy = 0,
(2ay + g) "'x + (-2ax + h)",y = ° and adding the operator ~.i to
it, we get
(2ay + g)~.iCPx + (-2ax + h)~.icpy = 0,
(2ay + g)~.iCPx + (-2ax + h)~.i"'y = 0.
(3.1)
(3.2)
(3.3)
Equations (3.2) and (3.3) mean the linear dependence of the lines
in the determinants
Therefore, the nonlinear summands in the Kadomtsev-Pogutse
equations vanish. Thus, the solutions of the Kadomtsev-Pogutse
equations invariant with respect
to the S = aC + hH + gG satisfy the linear system of the
equations
(2ay + g)cpx + (-2ax + h)cpy = 0,
(2ay + g)"'x + (-2ax + h)",y = 0, (3.4)
"', = CPz,
The first pair of these equations yields a dependence of cp and '"
on x and y
cp = cp(O, Z, t), '" = ",(0, z, t),
where O=(2ay+g)2+(-2ax+h)2. Taking this into account, the last pair
of equations (3.4) reduce to the form
"', = CPz,
One can easily integrate this system by changing the variables (= Z
+ t, 7) = Z - t. As a result, we get the following formulas for cp
and '"
cp = F(O, ()+<I>(O, 7))+(U,- U,,)ln O+(V,- V,,)
= F(O, ()+<I>(O, 7))+(U,+ U,,) In O+(V,+ V,,),
where
(3.5)
LAWS OF KADOMTSEV-POGUTSE EQUATIONS
~ = z + t, TJ = Z - t; U = U(~, TJ), v = V(~, TJ),
F( 0, ~) and cfJ( 0, ~) are arbitrary functions
dU u" = -, and so on
dTJ
41
Naturally, the arbitrary functions U, V, F, <I> entering into
(3.5) should be specialized in a proper way for selecting the
solutions, invariant with respect to one of the above-listed
subalgebras.
Class 2 Here we use the following general remark. Let L =
(<1>1, ... , <1>5) be a symmetries subalgebra of the
system OY. It is evident that the existence of the 2-invariant
solutions requires the compatibility of the system <1>1 = 0,
... , <l>s = 0 (if the latter is linear, then compatibility
means the existence of a nontrivial solution). Thus, the
incompatibility of the system <1>; = 0 means that all the
2-invariant solutions are exhausted by trivial ones.
Such a situation takes place for some of the subalgebras contained
in the list given in Section 2, namely .'l'Ll, 2Lz, 2tLl-2t1.3,
.'l'tZ.l, 2tz.z, .'l'L.3. Below, we will take the subalgebra 2 =
2tl.3 as a characteristic example.
The subalgebra 21.1.3 is generated as a linear space by elements of
the form
<1>1 = c+ /tF+2hzG-2gzH,
<l>z = M + fzF+ gzG+ hzH,
<1>3 = E+ /JF.
Hence, the system <1>1 = 0, <l>z = 0, <1>3 = 0
has the form
(y + 2hz)cpx - (x + 2gz)cpy + /tcp. = 0,
(y + 2hz)"'x - (x + 2gz)"'y + 11"'. = 0,
(x + gz)cpx + (y + hz)cpy + fzcp, - 2cp = 0,
(x + gz)"'x + (y + hz)cpy + fz"', - 2", = 0,
cpz + /Jcp, = 0,
"'z + /J"', =0,
and splits into equations containing either the function cp or the
function "'. We single out those containing cp:
(y + 2hz)CPx - (x + 2gz)cpx + I1CP, = 0,
(x + gz)cpx + (y + hz)cpy + Izcp, - zcp = 0,
cpz + /Jcp. = O.
It follows from the last equation that
rp = rp(X, y, 0), 0 = t- hz.
Then, rpt = rp8. Furthermore, solving the first pair of equations
of (3.6) with respect to rpx and rpy and writing down the
compatibility condition rpxy = rpyx , one gets the following
equation
[fz(h2x - g2Y) + !I(g2X + h2y + g~ + hm x
x rp8 + 2(g2Y - h2X)rp = O.
Its general solution is
_ ( ). ([f2(h2X - g2Y) + h2(g2 X + h2y + g2 + h2)]) rp - c x, y 0
2( h) . g2Y - 2 X
For brevity we denote the power of 0 by S. Let us substitute rp =
c(x, y) . OS into the first equation of (3.6)
(y + 2h2)(cxOS + c . OS In 0 . Sx) - (x + 2g2 ) x x (cyOS + cOs In
OSy) + IIcOS - 1 = O.
This equation contains an unknown function c(x, y), while it must
be valid for any o. But the left-hand part is a linear combination
of Os, OS In 0, OS-I with coefficients depending on x and y only.
Thus, the coefficients at Os, oS-I, OS In 0 must be identically
zero. In particular, a coefficient at oS-I, II . C = 0, whence, it
follows that c(x, y) = O. Hence, system (3.6) has only a zero
solution rp == O. The I{I == 0 identity is obtained
analogously.
Class 3 This case unites two sub algebras, IeL and IeI.2, from the
list in Section 2. Both of them contain as generators the
translations E and F by the variables z and t. Solutions invariant
with respect to these subalgebras, do not depend on z and t. This
makes it possible to reduce the Kadomtsev-Pogutse equations to the
form
rpxl{ly - rpyl{lx = 0,
rpxA1-rpy - rpyA1-rpx = I{Ix A1-l{Iy -l{IyA1-l{Ix.
The third generator of the subalgebra Iei.l yields two more
conditions
(1- 2m)rp + (y + mx + gl)rpx + (- x + my + h1)rpy = 0,
(I-2m)l{I+ mx + gl)l{Ix + (-x + my + h1)l{Iy = 0,
(3.7)
(3.8)
which are solved by the characteristic method. Together with the
first of Equations (3.7), it implies I{I = krp, k = const and
rp = rp( 0),
where 0 = O(x, y) is a known function whose explicit form may be
obtained from (3.8).
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 43
We note now that if the condition 0/ = k'P holds, then the first
equation of system (3.7) is automatically satisfied and the second
one reduces to the equation
(k 2 - l)[V"-'P, V "-d"-'P]z = o. The latter is the identity if k =
± 1 and reduces to the Liouville equation
studied in Section 5. The analogical deductions can be drawn with
respect to the subalgebra ,;t'i.2'
Class 4 Here we study the remaining three subalgebras from the list
in Section 2 which have not been included in the previous
considerations. They are 2:I3, 2: L, and
2:~,2' The system for finding 2:~.3-invariant solutions of the
Kadomtsev-Pogutse
equations is of the form
(x + gl)'Px + (y + hl)'Py,+ Z'Pz + ('P, = 0,
(x + gdo/x + (y + hl)o/y + zo/z + to/, = 0,
gzo/x + hzo/y + O/Z = 0,
gz'Px + hz'Py + 'Pz = 0,
g30/x + h30/Y + 0/, = 0,
0/, + [V"-'P, V"-o/Jz = 'Pz·
(3.9)
The first six equations of this system are linear and are solved by
the characteristic method:
Let
(3.10)
By substituting (3.10) into the Kadomtsev-Pogutse equations, we get
the follow ing system of two ordinary differential equations
((z + 1)(<1><1>'" - WW"') + ((z + 1)(g3( -
h3)(<1>'" - W''') +
+ ((z + 1)(<1>'<1>" - W'W") + (3g3(z -2h3 ( +
g3)(<I>" - W") +
+ 2((<1><1>' - WW') = 0,
<I>'W - <l>W' + (gz( - hz)(<I>' - W') -
gz<l> + g3 W = O.
(3.11)
44
fP = <I>«()(x - g2Z - g3 t + gl),
'" = 'I'(C)(x - g2Z - g3t+ gl),
where
C = y + h2 z - h3 t + hi, x- g2Z-