ñòð. 98 |

then its solutions can be looked for in the form [2]:

(29)

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

where ?(?) is an arbitrary smooth function, and what is more, functions ?(x) and

g(x) are to satisfy the following conditions:

?? ?g(x)

? µ (x) ? µ (x)

= 0, = ?(x).

?xµ ?xµ

To obtain all the P (1, 1) non-conjugated Ans?tze (29) we have to describe all the

a

inequivalent one-dimensional subalgebras of the Lie algebra AP (1, 1) spanned by the

operators (26) and (27) (see [2, 11]). In the paper we make use of a classification

adduced in [11]. Omitting cumbersome intermediate computations we give P (1, 1)

non-conjugated Ans?tze in the Table 1.

a

Table 1

N Algebra Invariant variable ? Ansatz

1? D ? J01 x0 + x1 u = ?(?) ? 2 ln(x0 ? x1 )

4

(1 + ?) ln(x1 ? x0 )?

2? D + ?J01 , ? = ?1 u = ?(?) ? ln(x0 + x1 )

? (1 ? ?) ln(x0 + x1 ) ?+1

ln(x0 ? x1 + 1/2)?

3? D ? J01 + P0 u = ?(?) ? 2 ln(x0 ? x1 + 1/2)

? 2(x0 + x1 )

4? 2 2

x0 ? x1 u = ?(?)

J01

5? u = ?(?)

P0 x1

6? P0 + P1 x0 ? x1 u = ?(?)

Symmetry reduction and exact solutions of nonlinear biwave equations 409

Remark. Inequivalent subalgebras adduced in the Table 1 are constructed by taking

into account an obvious fact that equation (7) is invariant under transformations of

the form:

x0 > x0 , x0 > x1 ,

and (30)

x1 > ?x1 ; x1 > x0 .

Substituting the Ans?tze obtained into the equation (7) we get the following

a

ordinary differential equations (ODE) for a function ?(?):

1? 0 = ?e? ,

? 2?

2? ?(4) (?2 ? 1)2 + 2?(3) ?(1 ? ?2 ) ? ?(2) (1 ? ?2 ) = exp ? + ,

16 ?+1

??

3? ?(4) ? ?(3) = e,

64

??

4? ?(4) ? 2 + 4?(3) ? + 2?(2) = e,

16

5? ?(4) = ?e? ,

6? 0 = ?e? .

Equation 5? has a particular solution

24

(? + c)?4 ,

? = ln ? > 0,

?

that leads to the following exact solutions of the equation (7):

24

(x0 + c1 )?4 ,

u = ln ? > 0,

?

(31)

24

(x1 + c2 )?4 ,

u = ln ? > 0.

?

Here c, c1 , c2 are arbitrary constants. This solutions are invariant under the

operators P0 and P1 accordingly.

In conclusion of the section let us note that the solutions (31) can be also obtained

by making use of the Ansatz in a Liouville form [2]:

? ?

2

? ?

? 24 ?1 (?1 )?2 (?2 ) ?

? ?

, ?1 = x0 + x1 , ?2 = x0 ? x1 ,

u = ln

? ? (?1 (?1 ) + ?2 (?2 ))4 ?

? ?

that reduces the equation (7) to one of the following systems:

1. ?1 = 0, ?2 = 0;

? ?

2?2 2?2

?1 ?2

2. ?1 =

? , ?2 =

? .

?1 ?2

Here ? and ? stand for the first and the second derivatives with respect to

? ?

a corresponding argument.

410 W.I. Fushchych, O.V. Roman, R.Z. Zhdanov

Integrating the above systems we get the following exact solutions of the equa-

tion (7):

(a2 ? b2 )2

24

(32)

u = ln ,

? (ax0 + bx1 + c)4

where a, b, c are arbitrary constants.

The solution (32) can be obtained from (31) by means of the final transformations

of the extended Poincar? group with generators (26) and (27).

e

5 Reduction and exact solutions

of the equation 22 u = ?uk

It follows from the Theorem 2 that the equation (6) with n = 1 is invariant under the

extended Poincar? group P (1, 1) with generators (26) and

e

? ? 4 ?

(33)

D = x0 + x1 + u.

1 ? k ?u

?x0 ?x1

If some equation admits a symmetry operator

? ?

X = ? µ (x) (34)

+ ?(x)u ,

?xµ ?u

then its solutions can be looked for in the form [2]:

(35)

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

provided functions ?(x) and f (x) satisfy the following system:

?? ?f (x)

? µ (x) ? µ (x) (36)

= 0, = ?(x)f (x).

?xµ ?xµ

A complete list of P (1, 1) non-conjugated Ans?tze invariant under the inequivalent

a

one-dimensional subalgebras of the algebra P (1, 1) is given in the Table 2.

Table 2

N Algebra Invariant variable ? Ansatz

2

1? D ? J01 x0 + x1 u = (x0 ? x1 ) 1?k ?(?)

4

??1

2? D + ?J01 , ? = ?1 (x0 ? x1 )(x0 + x1 ) ?+1 u = (x0 + x1 ) (1?k)(?+1) ?(?)

(x0 + x1 + 1 )?

2

3? 4

D + J01 + P0 u = exp (x1 ? x0 ) ?(?)

? exp 2(x1 ? x0 ) k?1

4? x2 ? x2 u = ?(?)

J01 0 1

?

5 u = ?(?)

P0 x1

6? P0 + P1 x0 + x1 u = ?(?)

Symmetry reduction and exact solutions of nonlinear biwave equations 411

Let us note that similar Ans?tze for the nonlinear wave equation

a

2u = ?uk , (37)

were obtained in [3].

Substituting the Ans?tze obtained into the equation (6) we get the following ODE

a

for a function ?(?):

1 + k (2) ?k

1? ?= ?,

(1 ? k)2 32

3k + 1

2? (? ? 1)2 ?(4) ? 2 + 2(? ? 1)(? + 1)2 + 2? ??(3) +

1?k

6? ? 10 8 ?

+ 2 ?2 ? 4? + 3 + ?(2) = (? + 1)2 ?k ,

+

1?k (1 ? k)2 16

5k ? 1 (3) 4k 2 ?k

3? ?(4) ? 2 + ?(2) =

? ?+ ?,

k?1 (1 ? k)2 64

?k

4? ?(4) ? 2 + 4?(3) ? + 2?(2) = ?,

16

5? ?(4) = ??k ,

6? ??k = 0.

Equations 1? , 2? , 4? have particular solutions of the form:

1

64 (k + 1)2 k?1

2

? ? k?1 , k = ?1

?=

? (k ? 1)4

and equation 5? has a particular solution of the form

1

8 (k + 1)(k + 3)(3k + 1) 1

k?1

4

? ? k?1 , k = ?1, ?3, ? ,

?=

(k ? 1)4

? 3

which lead to the following solutions of the equation (6):

1

2

? k?1

64 (k + 1)2 k?1

(x0 + x1 + c1 )(x0 ? x1 + c2 ) k = ?1,

u= ,

? (k ? 1)4

1

8 (k + 1)(k + 3)(3k + 1) 1

k?1

4

k = ?1, ?3, ? ,

u= (x0 + c3 ) 1?k ,

(k ? 1)4

? 3

1

8 (k + 1)(k + 3)(3k + 1) 1

k?1

4

k = ?1, ?3, ? ,

u= (x1 + c4 ) 1?k ,

(k ? 1)4

? 3

where c1 , c2 , c3 , c4 are arbitrary constants.

Note that equation (37) has analogous solutions (see e.g. [2]).

6 Reduction and exact solutions

of the equation 22 u = ?u?3

It follows from the Theorems 2, 3 that the equation

22 u = ?u?3 (38)

412 W.I. Fushchych, O.V. Roman, R.Z. Zhdanov

with n = 1 is invariant under the conformal group C(1, 1) with generators (26) and

? ? ?

D(1) = x0 + x1 +u ,

?x0 ?x1 ?u

(39)

?

= 2xµ D(1) ? (x? x? )

(1)

Kµ , µ, ? = 0, 1.

?xµ

By analogy with the preceding Section solutions of the equation (38) are looked

ñòð. 98 |