стр. 66 |

? x = x + V x0 , x0 = x0 ,

?

? x0 = e2s x0 , x = es x,

2

? ? (x ) = exp i(V x) ? i V x0 ? (x ) = e?3s/2 ?(x);

? ?(x);

? 2

?

? x0 = x0 /?, x = x/?,

?

ix 2 a

? ? (x ) = ? 1/2 exp ? = 1 ? ax0 .

? ?(x),

2?

3

xi ?i , ?2 = ? 2 , ?2 = ? 2 .

Здесь ? = (?1 , ?2 , ?3 ), ? = (?1 , ?2 , ?3 ), (x?) =

i=1

Указанные соотношения следует дополнить такими же преобразованиями для

переменной y.

1. Фущич В.И., Об одном способе исследования групповых свойств интегро-дифференциальных

уравнений, Укр. мат. журн., 1981, 33, № 6, 834–838.

2. Фущич В.И., Симметрия в уравнениях математической физики, в кн.: Теоретико-алгебраические

исследования в мат. физике, Киев, 1981, 134–138.

3. Estabrook F.В., Harrison В.К., Geometric approach to invariance groups, J. Math. Phys., 1971, 12,

№ 4, 653–666.

W.I. Fushchych, Scientific Works 2000, Vol. 2, 283–294.

The symmetry and some exact solutions

of the nonlinear many-dimensional Liouville,

d’Alembert and eikonal equations

W.I. FUSHCHYCH, N.I. SEROV

Multiparametrical exact solutions of the many-dimensional nonlinear d’Alembert, Li-

ouville, sine-Gordon and eikonal equations arc obtained. The maximally extensive local

invariance groups of the equations are determined and invariants of the extended Poi-

ncar? group are found.

e

1. Introduction

In 1881 Sophus Lie propounded to use the groups of continuous transformations

for finding the exact solutions of partial differential equations (PDE). Later on many

authors exploited Lie’s ideas to study PDE of mechanics and physics (see Ames [2],

Bluman and Cole [5], where a vast bibliography is cited and the historical aspects are

discussed).

The classical work of Birkhoff [4] is devoted to the construction of the exact

solutions of nonlinear hydrodynamics equations with the help of Lie’s methods. Birk-

hoff [4] was the first to formulate the group method to obtain similarity (automodel)

solutions of PDE. Many of the exact solutions have been obtained mainly for two-

dimensional PDE. Lately Ovsyannikov’s book [16] has dealt with the modern develop-

ment of Lie’s theory. Ovsyannikov formulated the method of finding the partly invari-

ant solutions of PDE. To find such solutions one has to enumerate all the non-

equivalent subgroups of the PDE invariance group. It is a very complicated problem.

For example, the five-dimensional d’Alembert equation invariance group has more

than 500 subgroups. Hence it is natural to seek more effective approaches for obtai-

ning the exact solutions of many-dimensional PDE admitting a wide invariance group.

The main ideas we use in our work arc closely connected with those of Birkhoff [4]

and Morgan [15]. The aim of our paper is to find the exact solutions of the following

nonlinear pde widely used in mathematical and theoretical physics:

pµ pµ u + ? exp u = 0, (1.1)

2u + ?uk = 0, (1.2)

pµ upµ u = 0, (1.3)

where pµ = ig µ? ?/?x? , gµ? is the metric tensor with the signature (+1, ?1, . . . , ?1),

pµ pµ = ?? 2 /?x2 + ? ? ?2, u = u(x), x = (x0 , x1 , . . . , xn?1 ), ?, k are arbitrary real

0

constants. We use the summation convention for the repeated indices.

Equation (1.2) plays a special role in the quantum field theory when k = 3 and

x = (x0 , . . . , x3 ): its solutions may be used to construct some solutions of the Yang–

Mills equation by virtue of the ’tHooft–Corrigan–Wilczek ansatz (see Actor [1]).

J. Phys. A: Math. Gen., 1983, 16, P. 3645–3656.

284 W.I. Fushchych, N.I. Serov

For the solutions of (1.1)–(1.3) we adopt the ansatz suggested by Fushchych [6]:

(1.4)

u(x) = ?(?)f (x) + g(x),

where ?(?) is an unknown function of the new variables ? = ?(x) = {?1 (x), . . .,

?n?1 (x)}, the number of which is one less than the number of variables x = (x0 , . . .,

xn?1 ). The new variables ?(x) and the functions f (x), g(x) are determined from the

Lagrange equations

dx0 dx1 dxn?1 du

= 1 = · · · = n?1 = (1.5)

,

?0 ? ? ?

where ? µ and ? are the functions from the infinitesimal invariance transformations

xµ = xµ + ?? µ (x, u) + O ?2 , u = u + ??(x, u) + O ?2 . (1.6)

If ? µ and ? have the form

? µ = ? µ (x), (1.7)

? = a(x)u + b(x),

it implies (1.4).

Having substituted (1.4) into (1.1)–(1.3) one obtains equations for ?(?) which are

often rather easy to solve.

2. The group properties of (1.1)–(1.3)

It is evident from the above, that to find the new variables ?(x) and the functions

f (x) and g(x) it is necessary to know the functions ? µ (x) and ?(x, u) explicitly. Hence

we shall study the group properties of (1.1)–(1.3).

Theorem 1. Equation (1.1) is invariant under the Poincar? group P (1, n ? 1) and

e

under the scale transformation group D(1). The basis elements of the corresponding

?

Lie algebra P (1, n ? 1) = {P (1, n ? 1), D(1)} have to form

? ?

Jµ? = xµ p? ? x? pµ , D = x? p? ? 2i

pµ = ig µ? (2.1)

, .

?x? ?u

?

Theorem 2. Equation (1.2) is invariant under the extended Poincar? group P (1, n ?

e

1), with basis elements of its Lie algebra having the form

? 2i ?

Jµ? = xµ p? ? x? pµ ,

pµ = ig µ? D = x? p? + (2.2)

, u.

1 ? k ?u

?x?

Theorem 3. Equation (1.3) admits the infinite-dimensional invariance group. The

infinitesimal operator of this group is as follows (we use Ovsyannikov’s [16] notati-

ons):

X = ? µ (x, u)?/?xµ + ?(x, u)?/?u,

(2.3)

? µ = ?bµ (u)x? x? + 2xµ b? (u)x? + cµ? x? + dµ (u), ? = ?(u),

where b? , cµ? , dµ , ? are arbitrary real functions of u and c0a = ca0 , cab = ?cba ,

c00 = c11 = · · · = cn?1 n?1 , a, b = 1, n ? 1.

Theorem 4. The equation

2u + F (x, u) = 0 (2.4)

The symmetry and some exact solutions 285

is invariant under the extended Poincar? group if and only if

e

(2.5)

F (x, u) = ?1 exp u,

or

F (x, u) = ?2 uk . (2.6)

where ?1 , ?2 , k are arbitrary constants, k = 1, the infinitesimal generators are given

in (2.1) and (2.2) respectively.

To prove these theorems one can use the Lie algorithm following e.g. Ovsyanni-

kov [16]. One can make sure that (2.4) with nonlinearities (2.5), (2.6) is invariant

?

under the group P (1, n ? 1) using final invariance transformations.

Note 1. Theorem 4 implies that there is only one equation of the form (2.4) with non-

?

polynomial nonlinearity invariant under P (1, n ? 1), and it is the Liouville equation.

Note 2. If n = 2, equation (1.1) admits the infinite-dimensional Lie group with the

generator X = ? 0 ?/?x0 + ? 1 ?/?x1 + ??/?u, where

? 0 = f (x0 + x1 ) + g(x0 ? x1 ), ? 1 = f (x0 + x1 ) ? g(x0 ? x1 ) + c1 ,

(2.7)

? = c2 ? ?? 0 /?x0 ,

f and g are arbitrary differentiable functions, c1 , c2 are constants.

Note 3. Equation (1.2) with n = 2 and ? = 0 is invariant under the infinite-

dimensional Lie group, as it takes place for the Liouville equation. The two-dimensio-

nal equation of gas dynamics has the same properties (see Fushchych and Serova [10]).

Apparently this property gives the possibility of finding the general solution of the

equations mentioned above.

?

3. The group P (1, 2) invariants

The question of finding of the invariants ?(x) is connected with the integration of

the Lagrange system (1,5). Generally speaking, equations (1.5) have infinitely many

solutions according to the various functions ? µ . Ovsyannikov [16] has proposed to

enumerate all the non-conjugate subgroups of the equations invariance group and to

integrate the system (1.5) for each subgroup. This way, as was previously mentioned,

is connected with the algebraic difficulties.

In this section we shall show the particular case of the group (1.6) for which the

system (1.5) is usually integrated.

Many fundamental equations of mathematical and theoretical physics are invariant

under the group IGL(n, R) of inhomogeneous linear transformations of n-dimensional

Minkowski space or under its subgroups, e.g. the Lorentz group, the Poincar? group,

e

the Galilei group etc.

It is well known that the functions ? µ for this group have the form

µ = 0, n ? 1,

? µ = cµ? x? + dµ , (3.1)

where cµ? and dµ are arbitrary constants.

Let us introduce the notations

dx0 dx1 dxn?1

= 1 = · · · = n?1 = dt. (3.2)

?0 ? ?

286 W.I. Fushchych, N.I. Serov

Using (3.1) and (3.2) one can write down the system (1.5) in the form

µ = 0, n ? 1.

xµ = cµ? x? + dµ , (3.3)

?

Equation (3.3) is a system of ordinary differential equations with constant coefficients

and it is well known how to find its general solution. After doing this one has to

eliinate the parameter to obtain the invariants ?(x).

?

If (1.1)–(1.3) are invariant under the group P (1, n ? 1), which is a subgroup of

?

IGL(n, R), we shall consider the determination of P (1, n ? 1) invariants in detail. For

simplicity we put n = 3. According to the conditions between the coefficients cµ? and

dµ in (3.1) we have obtained the following independent solutions of the system (1.5).

?2 = y? y ? (?? y ? )?2 ,

?1 = ?? y ? (?? y ? )a ,

(1)

where ?? ?? = ?? ? ? = 0, ?? ? ? = b = 0.

?1 = ?? y ? (?? y ? )?1 + ln ?? y ? , ?2 = y? y ? (?? y ? )?2 ,

(2)

where ?? ?? = ?? ? ? = 0, ?? ? ? = b = 0.

?1 = ln ?? y ? + b1 tan?1 [?? ? ? (?? y ? )?1 ], ?2 = y? y ? (?? y ? )?2 ,

(3)

where ?? ?? = b2 = 0, ?? ? ? = ?? ? ? = b3 = 0, ?? ? ? = ?? ? ? = ?? ? ? = 0, (?? y ? )2 +

(?? y ? )2 = b3 (?? x? )?2 (1 ? b2 ?2 ).

?2 = ?? z ? (?? y ? )?2 ,

?1 = ?? y ? + ln ?? y ? ,

(4)

where ?? ?? = ?? ? ? = ?? ?? = ?? ? ? = 0, ?? ? ? = b1 = 0, ?? ? ? = b2 = 0.

?1 = (?? y ? )2 + y? y ? , ?2 = ?? y ? + a ln ?? y ? ,

(5)

where ?? ?? = ?? ? ? = 0, ?? ? ? = ?1.

?2 = ?? y ? + a tan?1 [?? y ? (?? y ? )?1 ],

?1 = (?? y ? )2 ? y? y ? ,

(6)

where ?? ?? = ?? ? ? = b = 0, ?? ? ? = 1, ?? ? ? = ?? ? ? = ?? ? ? = 0, (?? y ? )2 +

(?? y ? )2 = b?1 .

1 1

(?? y ? )2 + a?? y ? , (?? y ? )3 + a?? y ? ?? y ? + a2 ?? y ? ,

(7) ?1 = ?2 =

2 2

where ?? ?? = ?? ? ? = ?? ? ? = 0, ?? ? ? = ??? ? ? = ?? ? ? = b = 0.

?1 = ?? x? , ?2 = x? x? , ?? ?? = b = 0.

(8)

?1 = (?? y ? )(?? y ? )?1 , ?2 = ?? y ? (?? y ? )?1 ,

(9)

where ?? ?? = a11 , ?? ? ? = a12 , . . ., ?? ? ? = a33 .

?1 = ?? x? , ?2 = ?? x? ,

(10)

where ?? ?? = ??? ? ? = 1, ?? ? ? = 0.

In these formulae y? = x? + a? , z? = x? + 1 a? , a? , ?? , ?? , ?? , a, b, bk , aik are

2

constants connected with the group parameters cµ? and dµ .

The symmetry and some exact solutions 287

To find f and g from (1.4) it is sufficient to integrate the equation

(3.4)

(du)/? = dt.

(3.4) yields

(3.5)

u(x) = ?(?) + g(x),

(3.6)

стр. 66 |