ñòð. 67 |

u01 = x1 + a[cu01 sin u11 + Du01 ],

c

u11 = x0 + a (u01 )2 ? C cos u11 + D .

2

The maximal local invariance group of equation (16) is the 7-parameter group. The

basic elements of the corresponding algebra are

P0 = ?x0 , P1 = ?x1 , P2 = ?u , D = x0 ?x0 + x1 ?x1 + 2u?u ,

(25)

Q1 = x1 ?u , Q2 = x2 ?u , Q3 = x1 x2 ?u .

It can be shown, that the system (24) has no solutions invariant under the operator

?0 P0 + ?1 P1 + ?2 P2 + dD + ?1 Q1 + ?2 Q2 + ?3 Q3 , where ?0 , ?1 , ?2 , d, ?1 , ?2 , ?3 are

arbitrary constants. Therefore, no solution of system (24) is invariant one for equation

(16).

Further, we consider the equation

(26)

[F (u)]1 = u22 .

We write the equation (26) in the form of the system

(27)

F (u) = ?x2 , ux2 = ?x1 .

Theorem 3. The system (27) is invariant with respect to the one-parameter Lie

group generated by an operator of the form

Q = ?x1 ?x1 + ??x2 + u?u (28)

1

if F = ln u .

Operator (28) is a nonlocal symmetry operator of equation (26). We use It to

construct the nonlocal ansatz and exact solutions of the equation

1

(29)

= u22 .

ln u 1

Nonlocal ansatzes for nonlinear wave equation 271

The ansatz

f (?) x2

(30)

u= , ?=

?(?) ? ln x1

x1

corresponds to operator (28), where f (?), ?(?) are unknown functions.

Substituting (30) into (27) we obtain the reduced system of ordinary differential

equations

(31)

?? + ? = ln f, f = ?.

The solution of system (31) has the form

?2 ?2 + 2c ?2 ?2

? + C ln(?2 + 2c) + c1 , (32)

?= ln f= + c.

2 2 2 2

1

Using the formula (30) and the substitution = z we obtain the solution of the

ln u

equation

11

(33)

z1 + e z z2 =0

z2 2

namely

?2

1

ez = + c,

2

(34)

x2

?= .

?2 2 +2c ?2

? + C ln(?2 + 2c) ? ln x1 + c1

ln ?

2 2 2

Formulae (34) give the solution of the nonlinear diffusion equation (33). In conclusion,

we emphasize that the finite transformations

z

x1 = e?a x1 , (35)

x2 = x2 + a?, z= , ? =0

1 + az

can be used for the nonlocal generating of solutions of equation (33), since the system

(27) admits the operator (28) in Lie sense. In this case the formula of generating

solutions takes the form

z (e?a x1 , x2 + a?)

(36)

z= ,

1 ? az (e?a x1 , x2 + a?)

where ? is the solution of the system

1 1

?x1 = ?zx2 (37)

ez , ?x2 = z ,

(z )2

z is the initial solution and z is the new solution of the equation (33), a is an

arbitrary constant.

Suggested approach can be effectively applied for the nonlocal generating of solu-

tions of equations which are invariant with respect to the group of contact transfor-

mations.

272 W.I. Fushchych, I.M. Tsyfra

1. Fushchych W., Shlelen W., Serov N., Symmetry analysis and exact solutions of nonlinear equations

of mathematical physics, Reidel, Kluwer Publ. Comp., 1993, 450 p.

2. Fuschchych W.I., Zhdanov R.Z., Nonlinear spinor equations: symmetry and exact solutions, Kèåâ,

Hayê. äyìêa, 1992, 288 p.

3. Fushchych W.I., How to expand symmetry of differential equations? in Symmetry and Solutions of

Nonlinear Equations of Mathematical Physics, Kèåâ, Èí-ò ìaòeìaòèêè, 1987, 4–16.

4. Fushchych W.I., On symmetry and exact solutions of many-dimentional nonlinear wave equations,

Óêð. ìàò. æóðí., 1987, 39, ¹ 1, 116–123.

5. Fushchych W.I., Tsifra I.M., On a reduction and solutions of nonlinear wave equations with broken

symmetry, J. Phys. A, 1987, 20, L45–L48.

6. Fushchych W.I., Serov M.I., Repeta V.C., Conditional symmetry reduction and exact solutions of

nonlinear wave equations, Äîïîâiäi AH Óêpa¿íè, 1991, ¹ 5, 29–34.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 273–282.

?

Nonlinear representations for Poincare

and Galilei algebras and nonlinear equations

for electromagnetic fields

W.I. FUSHCHYCH, I.M. TSYFRA, V.M. BOYKO

We construct nonlinear representations of the Poincar?, Galilei, and conformal algebras

e

on a set of the vector-functions ? = (E, H). A nonlinear complex equation of Euler

type for the electromagnetic field is proposed. The invariance algebra of this equation

is found.

1. Introduction

It is well known that the linear representations of the Poincar? algebra AP (1, 3)

e

and conformal algebra AC(1, 3), with the basis elements

Jµ? = xµ P? ? x? Pµ + Sµ? ,

Pµ = ig µ? ?? , (1)

D = x? P ? ? 2i, (2)

Kµ = 2xµ D ? (x? x? )Pµ + 2x? Sµ? , (3)

is realized on the set of solutions of the Maxwell equations for the electromagnetic

field in vacuum (see e.g. [1, 2])

?E ?H

= ?rot E, (4)

= rot H,

?t ?t

(5)

div E = 0, div H = 0.

Here Sµ? realize the representation D(0, 1) ? D(1, 0) of the Lorentz group.

Operators (1)–(3) satisfy the following commutation relations:

[Pµ , J?? ] = i(gµ? P? ? gµ? P? ), (6)

[Pµ , P? ] = 0,

[J?? , Jµ? ] = i(g?µ J?? + g?? J?µ ? g?µ J?? ? g?? J?µ ), (7)

[D, Pµ ] = ?iPµ , (8)

[D, Jµ? ] = 0,

[Kµ , P? ] = i(2J?µ ? 2gµ? D), [Kµ , J?? ] = i(gµ? K? ? gµ? K? ), (9)

[Kµ , D] = ?iKµ , (10)

[Kµ , K? ] = 0, µ, ?, ?, ? = 0, 1, 2, 3.

In this paper the nonlinear representations of the Poincar?, Galilei, and conformal

e

algebras for the electromagnetic field E, H are constructed. In particular, we prove

that the continuity equation for the electromagnetic field is not invariant under the

Lorentz group if the velocity of the electromagnetic field is taken in accordance with

J. Nonlinear Math. Phys., 1994, 1, ¹ 2, P. 210–221.

274 W.I. Fushchych, I.M. Tsyfra, V.M. Boyko

the Poynting definition. Conditional symmetry of the continuity equation is studied.

The complex Euler equation for the electromagnetic field is introduced. The symmetry

of this equation is investigated.

2. Formulation of the main results

The operators, realizing the nonlinear representations of the Poincar? algebras

e

AP (1, 3) = Pµ , Jµ? , AP1 (1, 3) = Pµ , Jµ? , D , and conformal algebra AC(1, 3) =

Pµ , Jµ? , D, Kµ , have the structure

(11)

Pµ = ?xµ ,

Jkl = xk ?xl ? xl ?xk + Skl , (12)

(13)

J0k = x0 ?xk + xk ?x0 + S0k , k, l = 1, 2, 3,

(14)

D = xµ ?xµ ,

K0 = x2 ?x0 + x0 xk ?xk + (xk ? x0 E k )?E k ? x0 H k ?H k , (15)

0

Kl = x0 xl ?x0 + xl xk ?xk + [xk E l ? x0 (E l E k ? H l H k )]?E k +

(16)

+ [xk H l ? x0 (H l E k + E l H k )]?H k ,

where

Skl = E k ?E l ? E l ?E k + H k ?H l ? H l ?H k ,

S0k = ?E k ? (E k E l ? H k H l )?E l ? (E k H l + H k E l )?H l .

The operators, realizing the nonlinear representations of the Galilei algebras

(2) (2) (2)

AG(2) (1, 3) = Pµ , Jkl , Gk , AG1 (1, 3) = Pµ , Jkl , Gk , D have the form:

Jkl = xk ?xl ? xl ?xk + Skl , (17)

Pµ = ?xµ ,

G2 = xk ?x0 ? (E k E l ? H k H l )?E l ? (E k H l + H k E l )?H l , (18)

k

D = x0 ?x0 + 2xk ?xk + E k ?E k + H k ?H k . (19)

We see by direct verification that all represented operators satisfy the commutation

relations of the algebras AP (1, 3), AC(1, 3), AG(1, 3).

3. Construction of nonlinear representations

In order to construct the nonlinear representations of Euclid-, Poincar?-, and Gali-

e

lei groups and their extensions the following idea was proposed in [2, 3]: to use

nonlinear equations invariant under these groups; it is necessary to find (point out,

guess) the equations, which admit symmetry operators having a nonlinear structure.

Such equation for the scalar field u(x0 , x1 , x2 , x3 ) is the eikonal equation

?u ?u

(20)

= 0, µ = 0, 1, 2, 3

?xµ ?xµ

which is invariant under the conformal algebra AC(1, 3) with the nonlinear operator

Kµ [2, 3].

Nonlinear representations for Poincar? and Galilei algebras

e 275

The nonlinear Euler equation for an ideal fluid

?vk ?vk

(21)

+ vl = 0, k = 1, 2, 3

?t ?xl

which is invariant under nonlinear representation of the AP (1, 3) algebra, with basis

elements

Jkl = xk ?xl ? xl ?xk + vk ?vl ? vl ?vk , (22)

Pµ = ?xµ ,

J0k = xk ?0 + x0 ?xk + ?vk ? vk vl ?vl , (23)

was proposed in [3] to construct the nonlinear representation for the vector field.

Note that equation (21) is also invariant with respect to the Galilei algebra AG(1, 3)

with the basis elements

Jkl = xk ?xl ? xl ?xk + vk ?vl ? vl ?vk , (24)

Pµ = ?xµ , Ga = x0 ?xa + ?va .

As mentioned in [2, 3] both the Lorentz–Poincar?–Einstein and Galilean principles

e

of relativity are valid for system (21). We use the following nonlinear system of

equations [4]

?E k k

?H k k

l ?E l ?H

(25)

+H = 0, +E = 0,

?x0 ?xl ?x0 ?xl

for constructing a nonlinear representation of the AP (1, 3) and AG(1, 3) algebras

for the electromagnetic field. To construct the basis elements of the AP (1, 3) and

AG(1, 3) algebras in explicit form we investigate the symmetry of system (25). We

search for the symmetry operators of equations (25) in the form:

X = ? µ ?xµ + ? l ?E l + ? l ?H l , (26)

where ? µ = ? µ (x, E, H), ? l = ? l (x, E, H), ? l = ? l (x, E, H).

Theorem 1. The maximal invariance algebra of system (25) in the class of operators

(26) is the 20-dimensional algebra, whose basis elements are given by the formulas

(27)

Pµ = ?xµ ,

(1)

Jkl = xk ?xl ? xl ?xk + E k ?E l ? E l ?E k + H k ?H l ? H l ?H k , (28)

(2)

ñòð. 67 |