ñòð. 19 |

heat equation using the operator u?u which is the counterpart of the other parabolic

symmetry operator L. Using M , we improve upon our result in that we are able to

include nonlinearities and still obtain a reduction to a parabolic equation. If we were

to use L instead, then we would reduce (1) to the heat equation with a complex

function. This, however, may be done only in the cases F = const = 0 and F = 0, as

it is only then that L appears as a symmetry. On writing ? = ueiw , one ?nds that

L = u?u whereas M = ?w . Therefore, equations admitting the symmetry M involve

only the derivatives of the phase.

In [17] we investigated equation (1) from a slightly di?erent point of view: taking

the phase-amplitude representation of ?, we used results about the compatibility of

the system

2v = F1 (v), ? µ v?µ v = F2 (v),

to obtain new solutions of non-Lie type (that is, not obtainable by reduction by Lie

symmetries). The same approach can be taken for the nonlinear Schr?dinger equation,

o

and the methods of [17] can also be combined with those of this article.

2.2. The Galilei ansatz and reduction to the Schr?dinger equation. Equa-

o

tion (1) is invariant under ?µ and M , and therefore under any constant linear combi-

nation of them:

?µ ?µ + kM. (3)

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

The operator (3) gives rise to the invariant surface conditions

? ?

?µ ?µ ? = ?ik ?

?µ ?µ ? = ik?,

?

for ? and ?, where ?µ and k are real constants. These conditions give us the Lagran-

gian system

?

dxµ d? d?

(4)

= = ?.

?ik ?

?µ ik?

It is straightforward to show that (4) is equivalent to

?

d(cx) d? d?

(5)

= = ?

?ik ?

c? ik?

for any constant four-vector c, where cx = cµ xµ , c? = cµ ?µ . Then choose ? light-like,

so that ?2 = 0 and, further, choose ?, ?, ? so that

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

That is, ?, ?, ?, ? is a hybrid 2+2 basis of Minkowski space consisting of two space-like

vectors (?, ?) and two light-like vectors (?, ?). Then put c in (5) successively equal to

?, ?, ?, ?, and we obtain the Lagrangian system

?

d(?x) d(?x) d(?x) d(?x) d? d?

(6)

= == = = = ?.

?ik ?

0 0 0 1 ik?

The system (6) then integrates to give

? = e?ik(?x) v (?x, ?x, ?x),

?

? = eik(?x) v(?x, ?x, ?x), (7)

?

where v is a smooth function. Substituting equations (7) as ansatzes in (1), we obtain

(after some elementary manipulation) the equation

?v 1 ?

?v ?

i = F (|v|)v,

?t 2k 2k

?2 ?2

where we have used the notation t = ?x, y1 = ?x, y2 = ?x and ? = 2. For

+

2

?y1 ?y2

convenience, we choose k = ? 2 , and we then have the nonlinear Schr?dinger equation

1

o

in 2+1 space-time dimensions

?v

= ??v + ?F (|v|)v. (8)

i

?t

This is a well-studied equation, at least in 1+1 space-time dimensions, exhibiting

soliton solutions and being completely integrable (possessing in?nitely many commu-

ting ?ows) for F (|v|) = |v|2 (see [18]). It has been studied in other dimensions in

[20–23, 27] in terms of symmetries and conditional symmetries.

The Cauchy problem for equation (8) is well-posed for t > 0, and (8) has solutions

which are singular for t = 0. This leads to similar problems for the wave equation

when ?x = 0, which is a characteristic (?2 = 0), and so the initial-value problem of (8)

is related to the initial-value problem of (1) on a characteristic, known as Goursat’s

problem. For the linear equation, this has been studied in [28].

Solutions of the relativistic nonlinear wave equation 85

It is an interesting question as to what quantum-mechanical implications (8) has

for (1), but we shall not pursue this in the present article.

We emphasise that the connection between the hyperbolic equation (1) and the

Schr?dinger equation (8) is obtained by an ansatz which reduces the number of space-

o

time dimensions by one; it is not a contraction as in [13].

2.3. Symmetries of the Schr?dinger equation (8). The symmetry algebra

o

of equation (8) is given by the following result: its classi?cation according to the

type of nonlinearity is in a direct correspondence to that of the symmetry algebra of

equation (1).

Theorem 2. Equation (8) has maximal point symmetry algebra (with the given vector

?elds as basis) depending on the nonlinearity F (|v|):

(i) AG2 (1, 2), when F (|v|) = const |v|2 :

Pa = ??a , J12 = x1 ?x2 ? x2 ?x1 ,

T = ?t ,

1

Ga = t?a + ixa (v?v ? v ?v ), D2 = 2t?t + xa ?a ? (v?v + v ?v ),

?? ??

2

1

S = t2 ?t + txa ?a + ixa xa (v?v ? v ?v ) ? t(v?v + v ?v ),

?? ??

4

1

M = ? i(v?v ? v ?v );

??

2

(ii) AG1 (1, 2), when F (|v|) = const |v|k , k = 0, 2:

1

Pa = ??a , J12 = x1 ?x2 ? x2 ?x1 , Ga = t?a + ixa (v?v ? v ?v ),

T = ?t , ??

2

1

D2 = 2t?t + xa ?a ? k (v?v + v ?v ), M = ? i(v?v ? v ?v );

2

?? ??

2

?

(iii) AG(1, 2), when F (|v|) = const |v|k , for any k but F = 0:

Pa = ?a , J12 = x1 ?x2 ? x2 ?x1 ,

T = ?t ,

1 1

Ga = t?a + ixa (v?v ? v ?v ), M = ? i(v?v ? v ?v );

?? ??

2 2

(iv) AG2 (1, 2) ? B , when F = 0, where B in?nite space of arbitrary solutions

of the free Schr?dinger equation:

o

1

J12 = x1 ?x2 ? x2 ?x1 , Ga = t?a + ixa (v?v ? v ?v ),

T = ?t , Pa = ? a , ??

2

1

S = t2 ?t + txa ?a + ixa xa (v?v ? v ?v ) ? t(v?v + v ?v ),

?? ??

4

1

M = ? i(v?v ? v ?v ), D = 2t?t + xa ?a , L = v?v + v ?v ,

?? ?? B?v ,

2

where B is an arbitrary solution of the free Schr?dinger equation.

o

The algebra in Theorem 2i is the Schr?dinger algebra [14], which is a subalgebra of

o

the conformal algebra. This is re?ected in the fact that the nonlinearity in Theorem 2i

is the same as in Theorem 1i, for which the wave equation (1) is invariant under the

conformal group. Note that Theorems 2iv, v correspond to Theorem 1v, since for

equation (8) the case F = const = 0 can be gauged to the case (iv) on putting

v = eit?F v, and then v satis?es the free (no potential) Schr?dinger equation. The

o

? ?

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

above is an exhaustive list of the types of symmetries for all the di?erent types of

nonlinearities. Again, in each of the four cases, we ?nd the operator M = i(v?v ? v ?v ),

??

and we can use this in a similar way to the reduction of the wave equation, in order to

reduce (8) to the corresponding Schr?dinger equation in 1+1 space-time dimensions;

o

this time with the same nonlinearity and ‘coupling’ constant ?. Thus we can think

of the linear and nonlinear Schr?dinger equations as part of a chain of successive

o

reductions, beginning with a nonlinear (hyperbolic) wave equation in n + 1 space-

time dimensions, as in (1).

Theorem 2 now allows us to classify the reductions of equation (8), according to

the type of nonlinearity. If we exclude the case F = 0, then there are only three types

of algebras: AG(1, 2) = T, Pa , Ga , J12 , M , AG1 (1, 2) = T, Pa , Ga , J12 , M, D , and

AG2 (1, 2) = T, Pa , Ga , J12 , M, D, S . These are the maximal symmetry algebras of

the equations:

?v ?

= ??v + ?F (|v|)v, F (|v|) = |v|k ,

with (9)

i F = 0,

?t

?v

= ??v + ?|v|k v, (10)

i k = 0, 2,

?t

?v

= ??v + ?|v|2 v, (11)

i

?t

respectively. The Lie algebra AG2 (1, 2) was considered in [19]. It is the semi-direct

sum

ASL(2, R) ? AO(2) + M, Pa , Ga ,

?

where ASL(2, R) is the Lie algebra of the group SL(2, R), and AO(2) is the Lie

algebra of the group O(2). The other two algebras are subalgebras of AG2 (1, 2).

3 Some exact solutions

In this section we obtain some exact solutions of the wave equation using results from

the tables in the appendix. The other reduced equations are di?cult to solve, so we

leave them for future consideration, remarking only that they give exact solutions of

equation (1) when we use the ansatz in equation (7).

First, we take the case of the subalgebra P2 , T + 2?M from Table 1 in the

appendix, with F (|?|) = |?|n and n > 0. The reduced equation is then

?

? + a? = ?|?|n ?.

On putting

?(?) = ?(?)ei?(?)

into this equation, with ?, ? being real functions and ? > 0, we obtain

?

? + a? ? ??2 = ??n+1 , (12)

?

? ?? (13)

?? + 2?? = 0.

Solutions of the relativistic nonlinear wave equation 87

Equation (13) readily integrates to give us

A

? (14)

? = 2,

?

where A is a constant of integration. Put now equation (14) into equation (12) and

we ?nd

A2

? + a? ? 3 = ??n+1 ,

?

?

which is the Ermakov–Pinney [31] equation when ? = 0. Multiplying this equation

by 2? and integrating, we obtain

?

A2 2? n+2

2 2

(15)

? + a? + 2 =

? ? + C,

? n+2

where C is another constant of integration. We now consider three cases of equa-

tion (15).

Case 1. A = 0, C = 0, a = 0. Since A = 0 here, we have ? = const, and (15)

becomes

2? n+2

? a?2 ,

?2 =

? ?

n+2

from which we deduce

d?

= ±? + C1 .

?

2? n+2

a?2

n+2 ?

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