ñòð. 18 |

P t = ?t , Pa = ?a ,

22

Jab = xa ?b ? xb ?a , J0 n+1 = t?t ? ??? ,

1 1

J0a = v xa ?t + (t + 2?)?xa + xa ?? ,

2

2

1 1

Ja n+1 = v ?xa ?t + (t ? 2?)?xa + xa ?? ,

2

2

n

D = ? t?t + xa ?a + ??? ? ?R ,

2

v 2

x2

x n

+ 2?2 ?? ? (t + 2?)?R ,

K0 = 2 t + ?t + (t + 2?)xa ?xa +

2 4 2

v x2 x2 n

Kn+1 = ? 2 t ? ?t + (t ? 2?)xa ?xa + ? 2?2 ?? ? (t ? 2?)?R ,

2 4 2

Ka = 2xa D ? (4t? ? x )?xa .

2

The above algebra is equivalent to the extended conformal algebra AC(1, n + 1) ?

N . In fact,with new variables

1 1

x0 = v (t + 2?), xn+1 = v (t ? 2?) (9)

2 2

the operators in Theorem 1 can be written as

J?? = x? ?? ? x? ?? , N = ?R ,

P ? = ?? ,

(10)

n

D = ?x? ?? + N, K? = ?x? D ? (xµ xµ )?? .

2

Exact solutions of system (7), (8) using symmetries have been given in [5] and

in [6]. Some examples of solutions are the following (we give the subalgebra, ansatz,

and the solutions):

A1 = J12 + dN, P3 + N, P4 (d ? 0)

Ansatz:

1 x1

? = ? t + f (?), R = x3 ? d arctan ? = x2 + x2 .

+ g(?), 1 2

2 x2

Solution:

x2 + x2

1

? =? t+? + C1 , ? = ±1,

1 2

2 2

x1 1

? ln(x2 + x2 ) + C2 ,

R = x3 + d arctan 1 2

x2 4

where C1 , C2 are constants.

A4 = J04 + dN, J23 + d2 N, P2 + P3

Ansatz:

1

? = f (?), R = d ln |t| + g(?), ? = x1 .

t

80 P. Basarab-Horwath, W.I. Fushchych

Solution:

(x1 + C1 )2 1

R = d ln |t| ? d + ln |x1 + C1 | + C2 .

?= ,

4t 2

A9 = J01 , J02 , J03 , J12 , J13 , J23

Ansatz:

x2 + x2 + x2

1 1

? = ? ? t.

f (?) + 1 2 3

?= , R = g(?),

4t 4t 2

Solution:

x2 ? 4C1 t + 8C1 x2 ? 2(t ? 2C1 )2

2

3

R = ? ln

?= , + C2 .

4t ? 8C1 t ? 2C1

2

A14 = J04 + a1 N, D + a2 N, P3 , (a1 , a2 arbitrary)

Ansatz:

x2 3 x1

R = g(?) + a1 ln |t| ? a1 + a2 + ln |x1 |,

? = 1 f (?), ?= .

t 2 x2

Solution:

x2 1

R = a1 ln |t| + a2 ? a1 + ln |x1 | ? 2(a2 + 1) ln |x2 | + C.

? = 1,

t 2

1. Basarab-Horwath P., Fushchych W., Barannyk L., J. Phys. A, 1995, 28, 5291.

2. Basarab-Horwath P., Fushchych W., Barannyk L., Rep. Math. Phys., 1997, 39, 353.

3. Basarab-Horwath P., Fushchych W., Roman O., Phys. Lett. A, 1997, 226, 150.

4. Roman O., Ph.D. Thesis, Kyiv, Institute of Mathematics, 1997.

5. Basarab-Horwath P., Fushchych W., Barannyk L.L., Some exact solutions of a conformally

invariant nonlinear Schr?dinger equation, Preprint LiTH-MATH-R 97-11, Link?ping University,

o o

1997.

6. Barannyk L.L., Ph.D. Thesis, Kyiv, Institute of Mathematics, 1997.

7. Feynman R.P., Phys. Rev., 1950, 80, 440.

8. Aghassi J.J., Roman P., Santilli R.P., Phys. Rev. D, 1970, 1, 2753; Nuovo Cimento, 1971, 5,

551.

9. Fushchych W., Seheda Yu., Ukr. Math. J., 1976, 28, 844.

10. Schiller R., Phys. Rev., 1962, 125, 1100.

11. Gu?ret Ph., Vigier J.P., Lett. Nuovo Cimento, 1983, 38, 125.

e

12. Guerra F., Pusterla M.R., Lett. Nuovo Cimento, 1982, 34, 351.

13. Holland P.R., The quantum theory of motion, Cambridge, 1995.

14. Basarab-Horwath P., Generalized hyperbolic wave equations and solutions obtained from relati-

vistic Schr?dinger equations, Preprint, Link?ping University, 1997.

o o

15. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 81–99.

Solutions of the relativistic nonlinear wave

equation by solutions of the nonlinear

Schr?dinger equation

o

P. BASARAB-HORWATH, W.I. FUSHCHYCH, L.F. BARANNYK

Using an ansatz for nonlinear complex wave equations obtained by using Lie point

symmetries, we show how to construct new solutions of the relativistic nonlinear wave

equation from those of a nonlinear Schr?dinger equation with the same nonlinearity.

o

This ansatz reduces the number of space-time variables by one, and is not related to a

contraction. We give some examples of other types of hyperbolic equations admitting

solutions based on nonlinear Schr?dinger equations.

o

1 Introduction

That nonlinear equations should play a role in quantum theory is not a new idea.

This idea was propagated by de Broglie, Iwanenko and Heisenberg [1–3]. Nonlinear

wave mechanics was taken up again by Bialynicki–Birula and Mycielski [4]. This

theme has also been of interest more recently [5], and much work on exact solutions

and modelling of nonlinear equations in quantum theory has also been done [12, 21,

22, 6].

In this article we consider a new aspect of some types of nonlinear relativistic

equations, and we obtain a connection between solutions of nonlinear Schr?dingero

equations and our nonlinear relativistic equations. Our starting point is the nonlinear

hyperbolic wave equation

2? + ?F (|?|)? = 0, (1)

where

?2 ?2 ?2 ?2

2= ? 2 ? 2 ? 2,

?x2 ?x1 ?x2 ?x3

0

with

?

xµ = gµ? x? , µ, ? = 0, . . . , 3, gµ? = g µ? = diag(1, ?1, ?1, ?1), |?| = (??)1/2 ,

?

and ? = ?(x0 , x1 , x2 , x3 ) is a complex function, ? being the complex conjugate of ?,

and we use summation over repeated indices (here and in the rest of the paper).

Using Lie point symmetries, exact solutions have been obtained for di?erent choices

of the nonlinearity F [7–12]. In this paper we obtain a new class of solutions to (1) by

using the symmetries of (1) to establish a connection between (1) and the nonlinear

Schr?dinger equation

o

?v

= ??v + ?F (|v|)v. (2)

i

??

Reports on Math. Phys., 1997, 39, ¹ 3, P. 353–374.

82 P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

Equation (2) is invariant under point transformations generated by the Gali-

lei group. Therefore it seems at ?rst surprising that a Poincar?-invariant equation

e

should be connected with a Galilei-invariant one. It is, however, known that the Poin-

car? algebra contains the Galilei algebra [20], and the conformal algebra contains the

e

Schr?dinger algebra [13–16]. The invariance of a restricted class of solutions of the

o

generalized Bhabha equations (invariant under the 1+4 Poincar? group) with respect

e

to the Galilei group was remarked upon in [20]. However, it is important to note that

equation (1) is not invariant under the Galilei group.

The novelty of our result is that we use a hitherto unexploited symmetry of (1)

to construct an ansatz (called the Galilei or parabolic ansatz) reducing (1) to (2), for

arbitrary nonlinearities in the right-hand side of (1). Thus, we show how nonlinear

equations themselves give rise to this connection. The ansatz we construct is shown

to work in other cases where the nonlinearity contains derivatives. This is explained

by the fact that the equations in question admit the same symmetry operator which

is crucial to the construction of the ansatz. Furthermore, we do not establish the

connection in terms of contractions, as is done in [13, 14].

The article is organized as follows: ?rst, we give a symmetry classi?cation of equa-

tion (1) and show how to construct the ansatz connecting (1) to (2). We also give the

symmetry classi?cation of (2), exhibiting the parallel with the symmetry classi?cation

of (1). We list the subalgebra classi?cation of the symmetry algebra of (2), together

with the corresponding ansatzes and reduced equations, in the appendix. Because of

the types of nonlinearity, we are able to solve only some of the reduced equations, in

Section 3. In Section 4, we give some examples of other equations for which our ansatz

works, and give solutions of the relativistic equations which are related to solitons of

the corresponding (using our reduction) Schr?dinger equations in 1+1 space-time

o

dimensions. We do not list exact solutions based on the heat equation: these can be

obtained by using the results of [19].

2 Symmetry and Galilei ansatz for equation (1)

2.1. Symmetry classi?cation. For the sake of completeness, we give the symmetry

classi?cation of equations of type (1) in the following result.

Theorem 1. The Lie point symmetry algebra of equation (1) has basis vector ?elds

as follows:

(i) when F (|?|) = const |?|2 :

??

?µ , Jµ? = xµ ?? ? x? ?µ , Kµ = 2xµ x? ?? ? x2 ?µ ? 2xµ (??? + ??? ),

?? ??

D = x? ?? ? (??? + ??? ), M = i(??? ? ??? ),

where x2 = xµ xµ and ?µ = ?/?xµ , ?? = ?/??;

(ii) when F (|?|) = const |?|k , k = 0, 2:

Jµ? = xµ ?? ? x? ?µ ,

?µ ,

2 ?? ??

= x? ?? ? (??? + ??? ), M = i(??? ? ??? );

D(k)

k

?

(iii) when F (|?|) = const |?|k for any k, but F = 0:

??

Jµ? = xµ ?? ? x? ?µ , M = i(??? ? ??? );

?µ ,

Solutions of the relativistic nonlinear wave equation 83

(iv) when F (|?|) = const = 0:

??

?µ , Jµ? = xµ ?? ? x? ?µ , M = i(??? ? ??? ),

?? ? ?

L = ??? + ??? , L1 = i(??? ? ??? ), L2 = ??? + ??? , B?? ,

? ?

where B is an arbitrary solution of 2? = const ?;

(v) when F (|?|) = 0:

??

?µ , Jµ? = xµ ?? ? x? ?µ , Kµ = 2xµ x? ?? ? x2 ?µ ? 2xµ (??? + ??? ),

?? ??

D = xµ ?µ , M = i(??? ? ??? ), L = ??? + ??? ,

? ?

L1 = i(??? ? ??? ), L2 = ??? + ??? , B?? ,

? ?

where B is an arbitrary solution of 2? = 0.

The ?rst case, F (|?|) = |?|2 , gives us the extended conformal algebra, the second

case gives the extended Poincar? algebra. In all ?ve cases (which exhaust all possible

e

nonlinearities of the given type), the symmetry algebra contains the subalgebra Pµ ,

??

Jµ? , which is the Poincar? algebra, and the operator M = i(??? ? ??? ). It is

e

this operator which we combine with the generators of space-time translations ?µ

in order to build an ansatz which reduces equation (1) to a nonlinear Schr?dinger

o

equation. This gives a reduction of a hyperbolic equation to a parabolic equation,

and for this reason we call it a parabolic symmetry of the nonlinear wave equation.

In this fashion we are able to construct new solutions of (1), even making a contact

with the Zakharov–Shabat soliton solution [18] when F (|?|) = |?|2 . The appearance

of the parabolic symmetry M is a feature of the fact that ? is a complex-valued

function and of the type of nonlinearity we consider. In our previous article [19] we

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