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W.I. Fushchych, Scientific Works 2001, Vol. 3, 251–255.

On a reduction and solutions of non-linear

wave equations with broken symmetry

W.I. FUSHCHYCH, I.M. TSYFRA

A generalised definition for invariance of partial differential equations is proposed. Exact

solutions of the equations with broken symmetry are obtained.

Let us consider the non-linear wave equation

2u + F1 (u) = 0, u = u(x0 , x1 , x2 , x3 ),

(1)

2 = ?0 ? ? 1 ? ? 2 ? ? 3 ,

2 2 2 2

?µ = ?/?xµ , µ = 0, 1, 2, 3,

where F1 (u) is an arbitrary smooth function. The ansatz

(2)

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

suggested by Fushchych [5] was used to construct the family of exact solutions of

equations (1). f (x), g(x) are given functions, ?(?) is the function to be determined

and ? = (?1 , ?2 , ?3 ) are new invariant variable. Wide classes of exact solutions of

equation (1) have been constructed by Fushchych and Serov [7, 8], Fushchych et al

[10] and Fushchych and Shtelen [9]. It is important to note that Poincar? invariance

e

of equation (1) was used.

The possibility of using an ansatz of type (2) to find exact solutions of the non-

linear wave equations with broken symmetry naturally arises in connection with the

fact that many equations of theoretical physics are not invariant with respect to the

Poincar?, Galilei and Euclidean groups. A more specific formulation of this problem

e

is as follows: are we able to construct the solutions of wave equations not invariant

with respect to the Lorentz groups, for example, but nevertheless with the help of

the Lorentz-invariant ansatz?

The present letter suggests an affirmative answer to this question, i.e. we construct

the many-dimensional non-linear wave equations with broken symmetry. The multi-

parametrical exact solutions of these equations are found with the help of ansatz (2),

previously used to find exact solutions of Poincar?- and Galilei-invariant equations

e

only. It is obvious that ansatz (2) cannot be applied to the equations with arbitrary

breakdown of symmetry, which is why the equation with the breakdown of symmetry

should have some hidden symmetry. The set of equations with such symmetry was

considered by Fushchych and Nikitin [6]. We do not deal with the symmetry of all

the solutions of the equations but only with a definite subset of solutions, which may

be much wider that the symmetry of the equation itself. This idea will be used below.

Let us consider the wave equation with broken symmetry

Lu ? 2u + F (x, u, u) = 0, (3)

1

J. Phys. A: Math. Gen., 1987, 20, L45–L48.

252 W.I. Fushchych, I.M. Tsyfra

where F (x, u, u) is an arbitrary smooth function, depending on x = (x0 , x1 , x2 , x3 ),

1

u ? (?u/?x0 , ?u/?x1 , ?u/?x2 , ?u/?x3 ). Following Fushchych [3] we generalise the

1

Lie definition of invariance of equation (3).

Definition. We shall say that equation (3) is invariant with respect to some set of

? ?

operators Q = {QA }, A = 1, 2, . . . , N , a number of linearly independent operators,

if the following condition is fulfilled:

? (4)

QA Lu = 0,

Lu = 0,

?

{QA u} = 0

?

where {QA u} = 0 is a set of equations

? ? ? ?

D2 QA u = 0, Dn QA u = 0, (5)

QA u = 0, DQA u = 0, ...,

where D is an operator of total differentiation. Condition (4) is a necessary condition

for reduction of differential equations.

Definition (4) is a generalisation of the Lie definition (see, e.g., Ovsyannikov [12])

? (6)

QA Lu = 0,

Lu=0

?

where QA are a number of first-order differential operators forming a Lie algebra.

To demonstrate the efficacy of definition (4) and to find exact solutions of equation

(3) we choose the function F in a form

2 2 2 2

?0 ?u ?1 ?u

F =? + +

x0 ?x0 x1 ?x1

(7)

2 2 2 2

?2 ?u ?3 ?u

+ + ,

x2 ?x2 x3 ?x3

where ?µ are arbitrary parameters and xµ = 0.

Theorem. The maximal local (in the Lie sense) invariance group of equations (3)

and (7) is the two-parametrical group of the transformations

xµ = ea xµ , u = e2a u (8)

and

c = constant,

u = u + c,

where a is real parameter.

The proof of the theorem is reduced to application of the well known Lie algorithm

and we do not present it here. One can make sure non-linearity breaks the rotational

and translational symmetry.

Now we show that the Lorentz-non-invariant equations (3) and (7) are reduced to

an ordinary differential equation with the help of the Lorentz-invariant ansatz

? = xµ xµ = x2 ? x2 ? x2 ? x2 . (9)

u = ?(?), 0 1 2 3

Reduction and solutions of equations with broken symmetry 253

Substituting (9) into (3) and (7) we obtain the ordinary differential equation

2

d2 ? d? d?

= ??2 ?2 = ? 2 ? ? 2 ? ? 2 ? ? 2 . (10)

? 2 +2 , 0 1 2 3

d? d? d?

Solving equation (10), we obtain

?1/2 ?1/2

tan?1 ? ??2

?(?) = 2 ??2 ??2 > 0, (11)

,

1/2

?2 +?

2 ?1/2

?(?) = ? ? ??2 < 0. (12)

ln ,

1/2

??

(?2 )

Thus the Lorentz-non-invariant (in the Lie sense) equations (3) and (7) are reduced

to an ordinary differential equation.

Formulae (11) and (12) give a Lorentz-invariant family of solutions of equations

(3) and (7). It means that the following set of conditions is fulfilled:

(13)

Jµ? u(x) = 0, µ, ? = 0, 1, 2, 3,

? ? ? ?

? xb (14)

J0a = x0 + xa , Jab = xa , a, b = 1, 2, 3

?xa ?x0 ?xb ?xa

for the set of solutions (11) and (12).

The operators (14) generate Lorentz transformations. Equations (13) are the con-

crete realisation of the first equation of (5). In this case the index A varies from 1 to

6.

Thus, equations (13) pick out a Lorentz-invariant subset of the set of all solutions

of equations (3) and (7). In other words, equations (3) and (7) are Lorentz-invariant

in the sense of definition (4).

Now let us consider the equation

?2u 12

= ??u(?u)2 , (15)

?= m.

?t2 3

It is simple to verify that equation (15) is not invariant with respect to Galilean

transformations, generated by operators

?

(16)

Ga = t + mxa , a = 1, 2, 3.

?xa

?

In this case equations {QA u} = 0 are

?u

? mxa u = 0, (17)

Ga u = t

?xa

?

(18)

(Ga u) = 0.

?t

Thus equation (15) is invariant under transformations generated by operators (16)

in the sense of definition (4). It means that the subset of solutions of equations

(15) picked out by means of conditions (17) and (18) is invariant under Galilean

transformations while equation (15) is not invariant under these transformations.

254 W.I. Fushchych, I.M. Tsyfra

The Galilean-invariant ansatz has the form

u = ?(t) + m x2 + x2 + x2 /2t, (19)

? = t, f = 1.

1 2 3

Substituting (19) into (15), we obtain

d2 ?

- u = m x2 + x2 + x2 /2t + At + C, (20)

=0 1 2 3

?t2

where A and C are arbitrary constants.

A generalised definition of the invariance (4) can be applied to the system of partial

differential equations.

Let us consider, for example, a non-linear Dirac system of equations:

?µ ? µ ? + g 2?(xµ ? µ )? ? x2 /c? x? ?(cµ ? µ )? M ?1 (x)(??)1/3 ? = 0,

? ? ?

M (x) = 2(c? x? )?1 ?Sµ? cµ ? ? ? + ??,

? ?

(21)

1

i(?µ ?? ? ?? ?µ )

Sµ? = µ, ?, ? = 0, 1, 2, 3,

4

where g, ?µ , c? are arbitrary parameters.

Equation (21) is not invariant under conformal transformations. Nevertheless, it is

reduced to the system of ordinary differential equations

d?

i?µ ? µ + g(??)1/3 ? = 0 (22)

?

d?

with the help of the conformally invariant ansatz (4)

?(x) = ?µ xµ /(x2 )2 ?(?), ? = ?µ xµ /x2 , ? 2 = 0, x2 = xµ xµ = 0, (23)

where ?(?) is the four-component spinor depending on a variable ?. The general

solution of equation (22) is the vector function

?µ ? µ

? = exp ?i g(??)1/3 ? ?, (24)

?

2

?

where ? is a constant spinor.

Equation (21) is invariant under the transformations generated by the operator

cµ K µ on a set of solutions of the equations

cµ K µ ? = 0,

(25)

cµ Kµ = 2(cx)(x?) ? x2 (c?) + 2(cx) ? (?c)(?x).

In conclusion we note that an idea like the one set forward here was used by

Bluman and Cole [2], Ames [1], Fokas [3] and Olver and Rosenau [11], as was kindly

indicated by the referee.

We are grateful to the referee for his valuable remarks.

1. Ames W.F., Nonlinear partial differential equations in engineering, Vol.2, New York, Academic,

1972.

2. Bluman G.W., Cole I.D., J. Math. Mech., 1969, 18, 1025–1042.

3. Fokas A.S., PhD thesis, California Institute of Technology, 1979.

Reduction and solutions of equations with broken symmetry 255

4. Fushchych W.I., Dokl. Akad. Nauk, 1979, 246, 846–850.

5. Fushchych W.I., The symmetry of mathematical physics problems, in Algebraic-Theoretical Studies

in Mathematical Physics, Kiev, Mathematical Institute, 1981, 6–28.

6. Fushchych W.I., Nikitin A.G., Symmetry of Maxwell Equations, Kiev, Naukova Dumka, 1983.

7. Fushchych W.I., Serov N.I., J. Phys. A: Math Gen., 1983, 16, 3645–3656.

8. Fushchych W.I., Serov N.I., Dokl. Akad. Nauk, 1983, 273, 543.

9. Fushchych W.I., Shtelen W.M., J. Phys. A: Math Gen., 1983, 16, 271–277.

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