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Bµ (? )xµ + f2 (? ) = 0 (20)

and Aµ , Bµ , f1 , f2 are arbitrary functions of one variable satisfying

?

Aµ Aµ = ?1, Aµ B µ = 0, Aµ B µ = 0, Bµ B µ = 0. (21)

The abtive theorems give us some general information about the solutions of (18)

in particular cases. Of course, these results express the solution in terms of other

functions, but now we know how to generate these functions: we have to choose

Aµ , Bµ , f1 , f2 appropriately so as to define both ? and then w (as we do below

in a particular case). In this way, we have a systematic way of obtaining solutions

of (18). Our approach to the solution of the nonlinear d’Alembert equation is based

on a change of variable as in (2), and a decomposition of the equation (4) for the

new variable w into ‘component’ equations (5), (6) and (7). Equation (5) involves

the change of variable and the nonlinearity F (u), whereas (7) provides us with a

system which can be dealt with using theorems 1–3. The d’Alembert equation with

nonlinearity sin u was discussed in [4], where a change of variable together with a

decomposition of the ensuing equation was also used. The result obtained in [4] can

be obtained with our results, as follows. In (20), (21) put

Aµ = ?µ , Bµ = ?µ ,

where ?µ = ?µ ? ?µ , ?µ , ?µ , ?µ are constant vectors satisfying

?µ ?µ = ??µ ? µ = ??µ ? µ = 1, ?µ ? µ = ?µ ? µ = ?µ ? µ = 0.

Equation (20) then defines ? through

f2 (? ) = ??µ xµ

412 P. Basarab-Horwath, W.I. Fushchych, M. Serov

and on choosing f2 invertible we obtain the solution of (19)

w = ?µ xµ + f (?µ xµ ).

When F (u) = ? sin u we obtain

u = 4 arctan exp[?µ xµ + f (?µ xµ )], (22)

where f is an arbitrary differentiable function. Equation (22) is the solution obtained

in [4]. As can be seen, our method gives a useful way of obtaining exact solutions of

nonlinear equations.

WF acknowledges support by the Swedish Natural Sciences Research Council,

grant number R-RA 9423-307.

1. Fushchych W.I., Zhdanov R.Z., Revenko I.V., General solutions of the nonlinear wave equation and

the eikonal equation, Ukr. Math. J., 1991, 43, 1420–1460 (in Russian).

2. Fushchych W.I., Zhdanov R.Z., Revenko I.V., Compatibility and solutions of the nonlinear d’Alem-

bert and Hamilton equations, Preprint, Institute of Mathematics, Ukrainian Academy of Sciences,

Kiev, 1990 (in Russian).

3. Fushchych W.I., Zhdanov R.Z., Yehorchenko I.A., On the reduction of nonlinear multidimensional

wave equations and compatibility of the d’Alembert–Hamilton system, J. Math. Anal. Appl., 1991,

161, 352–361.

4. Ouroushev D., Martinov N., Grigorov A., An approach for solving the two-dimensional sine-Gordon

equation, J. Phys. A: Math. Gen., 1991, 24, L527–L528.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 413–414.

Symmetry analysis. Preface

W.I. FUSHCHYCH

Till now there are no general methods of investigation of arbitrary nonlinear partial

differential equations (PDE). However if a nonlinear equation is beautiful, that is to

say it possesses a non-trivial symmetry, than it is possible to obtain rather wide

and rich information about its solutions; to carry out reduction of multidimensional

equations to ordinary differential equations [1], to construct classes of exact and

approximate solutions, investigate asymptotic of special classes of solutions, etc. [1–

6].

It is important to point out that beautiful equations are these ones which are

widely used in mathematical and theoretical physics, and in applied mathematics. It is

connected with the fact that mathematical models of real processes must be of such a

form that e.g. conservation laws of energy, momentum, angular momentum of motion

or relativity principles [1, 3] and other important principles of physics are satisfied in

these models. The beauty (or symmetry, or approximate symmetry) of an equation has

a lot of forms: local (or Lie, as we call it in [2, 6]), nonlocal (non-Lie [2]), discrete,

non-group, non-algebraic. Therefore it is not simple to give a mathematically correct,

effective and general enough definition of beauty of an equation.

The principal ideas and methods of investigation of group (local) properties of

partial differential equations PDE are developed by Sophus Lie. These methods enable

to study group properties of an arbitrary partial differential equation. The above

methods form a part of modern theory of differential equations called on L.V. Ov-

syannikov and N.Kh. Ibragimov suggestion “Group analysis of PDE” [4, 5].

Since PDE prove rather often to possess symmetry that cannot be presented in

terms of Lie groups or Lie algebras, we use a more general term “Symmetry analysis”

suggested in [2, 6]. Symmetry analysis is the aggregate of mathematical methods for

investigating local, geometry, non-geometry, discrete, inner and dynamic symmetries

of PDE.

The present collection contains papers by participants of the Seminar “Symmetry

analysis of mathematical physics equations” (Institute of Mathematics of the Academy

of Sciences of Ukraine) in which two scientific directions are considered:

1. Conditional symmetry of equations of nonlinear mathematical physics.

2. Local, non-local symmetry and construction of classes of exact solutions of

nonlinear PDE.

In conclusion, I adduce several beautiful second-order PDE, that have not been

investigated yet

a(u, ?, 2u) + b(u, ?, u? u? u?? ) + c(u, ?, det uµ? ) = F (u, ?),

2 2 2

?u ?u ?u

? ? u? u = ? ? ··· ?

?

,

?x0 ?x1 ?xn

in Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute Mathematics, 1992, P. 5–

6.

414 W.I. Fushchych

?u ?u ? 2 u

u? u? u?? =

u = u(x0 , x1 , . . . , xn ), ,

?x? ?x? ?x? ?x?

?2u

det uµ? ? det ;

?xµ ?x?

?2E ?2H ?E ?H

? v 2 ?E = 0, ? v 2 ?H = 0, v = v E, H, , ;

2 2

?t ?t ?x ?x

?vµ

v 2 ? v? v ? = v 0 ? v 1 ? v 2 ? v 3 ;

2 2 2 3

v? = 0,

?x?

? ?

+ vk vl = Fl (v1 , v2 , v3 ), l = 1, 2, 3;

?t ?xk

?2u ?2u

= F (u, ?, 2u);

?xµ ?x? ?xµ ?x?

? ?u ? ? ?u ?

+ + u = F (u).

?t ?xb ?xb ?t ?xa ?xa

In the above formulae a, b, c, F , F1 arbitrary smooth

1. Fushchych W., Shtelen V., Serov M., Symmetry and exact solutions of equations of nonlinear

mathematical physics, Kiev, Naukova Dumka, 1989 (in Russian).

2. Fushchych W.I., A new method of study of the group properties, Dokl. AN USSR, 1979, 246, ¹ 4,

846–850.

3. Fushchych W.I., Nikitin A.G., Symmetry of equations of quantum mechanics, Moscow, Nauka, 1990

(in Russian).

4. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.

5. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.

6. Fushchych W.I., On symmetry and exact solutions of multidimensional nonlinear wave equations,

Ukr. Math. J., 1987, 39, ¹ 1, 116–123.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 415–431.

Conditional symmetry of equations

of nonlinear mathematical physics

W.I. FUSHCHYCH

1. Introduction. In this paper we present some results on conditional symmetry

of nonlinear equations of mathematical and theoretical physics which were obtained

in the Institute of Mathematics of Ukrainian Academy of Science. The term and the

concept of “conditional symmetry of equation” or “conditional invariance” had been

introduced in [1–10].

Speaking about the conditional symmetry of an equation, we mean of the symmet-

ry of some subset of its solutions. To be a constructive one, such general definition

needs, some more details. To study conditional symmetry means to give analytical

description of conditions (constraints) for the set of solutions of an equation under

study picking out subsets having wider (or another) symmetry properties than the

whole set of solutions. Having carried out such description one can obtain solutions

which cannot be obtained within the framework of the classical Vie approach (as it

is known, in the Lie approach reduction, of the multi-dimensional partial differential

equation (PDE) to equations with less number of independent variables is carried out

by means of symmetry of the set of its solutions in a whole).

Euler, Bateman, Lie, Smirnov and Sobolev (1932) and many other classics used

implicitly symmetry of subsets of solutions for linear d’Alembert and Laplace equa-

tions to construct their exact solutions. Not long ago Bluman and Cole [11] suggested

the “non-classical method of solutions invariant under group” for the linear heat

equation. Olver and Rosenau (1986) [12] constructed solutions of the one-dimensional

nonlinear acoustics equation

u00 = ? 2 u/?t2 , u11 = ? 2 u/?x2 (1)

u00 = uu11 ,

which cannot be obtained by means of Lie method. Clarkson and Kruskal suggested

“new method of invariant reduction of the Boussinesq equation”

1

(2)

u00 + u11 + u1111 = 0.

2

Conclusion 1. Using the concept of “conditional symmetry of PDE” we can obtain

the above results within the framework of the unified symmetry approach.

Conclusion 2. The majority of linear and nonlinear equations of mathematical

physics: d’Alembert, Maxwell, Schr?dinger, Dirac, heat, acoustics, KdV equations

o

possess some conditional symmetry.

Note 1. All solutions of the Boussinesq equation (2) constructed by Clarkson and

Kruskal had been obtained independently by Levi and Winternitz [14], and by Fu-

shchych and Serov [10], using the concept of conditional symmetry.

in Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute Mathematics, 1992, P. 7–

27.

416 W.I. Fushchych

Let us consider some PDE

L(x, u, u, u, . . . , u) = 0, (3)

s

12

where u = u(x), x ? R(n + 1), u(x) ? R, u is the set of s-th order partial derivatives

s

of u(x).

According to Lie, the equation (3) is invariant under the first-order differential

operator

? ?

X = ? µ (x, u) (4)

+ ?(x, u)

?xµ ?u

if the following condition is satisfied:

? (5)

X L = ?L XL = 0,

s s L=0

where X is the s-th prolongation of the operator X, ? = ?(x, u, u, u, . . . , u) is some

s

12

s

differential expression.

Let us designate by the symbol Q = {Q, . . . , Q} a collection of operators not

1 k

belonging to the invariance algebra (IA) of the equation (3), i.e. Q ? IA.

Definition 1 [2, 5]. We say that the equation (3) is conditionally-invariant under

the operators Q if there exists some additional condition

L1 (x, u, u, u, . . . , u) = 0 (6)

s

12

to be compatible.

The additional condition (6) picks out some subset from the whole set of solutions

of the equation (3). It appears that for many important equations of mathematical

physics such subsets admit the wider symmetry than the whole set of solutions. Such

subsets are to be constructed.

Let the operator Q act on the equation (3) as follows:

(7)

Q L = ?0 L + ?1 L1

s

or

QL = 0,

Lu = 0

s L1 u = 0

where ?0 , ?1 are some differential expressions depending on x, u, u, u, . . . , u, Q is the

s

12 s

s-th prolongation of the operator Q. Then the invariance condition reads

(8)

Q L1 = ?2 L + ?3 L1 ,

s

where ?2 , ?3 are some differential expressions.

The principal problem of our approach is to describe in explicit form equations of

the form (6) which extend symmetry of the equation (6).

The principal and difficult problem can be essentially simplified if one chooses the

following nonlinear first-order PDE as an additional condition (6):

(9)

Qu = 0,

Conditional symmetry of equations of nonlinear mathematical physics 417

where

?µ ? ?/?xµ , ?u ? ?/?u.

Q = j µ (xµ , u)?µ + z(xµ , u)?u , (10)

In this case, the invariance condition for the system of equations (3), (9) takes the

form

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