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?x0 n

1

is fundamental to mathematical physics: it describes spinless mesons when n = 3,

and is the paradigm of a hyperbolic equation. Its symmetry properties are also known

[1, 2], and one has the following result concerning the Lie point symmetries of (1):

Proposition 1. The maximal Lie point symmetry algebra of equation (1) has basis

Jµ? = xµ ?? ? x? ?µ (2)

P µ = ?µ , I = u?u ,

when m = 0 and

Pµ = ?µ , I = u?u , Jµ? = xµ ?? ? x? ?µ ,

(3)

D = xµ ?µ , Kµ = 2xµ D ? x2 ?µ ? 2xµ u?u

when m = 0, where

? ?

, xµ = gµ? x? ,

?u = , ?µ =

?xµ

?u

= diag (1, ?1, . . . , ?1), µ, ? = 0, 1, 2, . . . , n.

gµ?

The symmetries can be used to build ansatzes for exact solutions of (1), which then

reduce the equation to a partial differential equation with fewer independent variables

or even to an ordinary differential equation [1, 2]. These ansatzes and reductions

are based on a subalgebra analysis of parts of the symmetry algebra. The reduced

equations do not always have nice symmetry properties, so that a full analysis of the

resulting equations has not been carried out to this date. In this article we study

a reduction which, as far as we know, has not been done before, and which links up

solutions of the wave equation (1) in R(1, n) with those of the linear heat equation in

R(1, n?1). We consider equation (1) with real u: the complex case with nonlinearities

is studied in [3].

In [1, 2, 4], the reduction of the nonlinear wave equation

2u = F (u) (1a)

J. Phys. A: Math. Gen., 1995, 28, P. 5291–5304.

312 P. Basarab-Horwath, L. Barannyk, W.I. Fushchych

is considered and its reduction (to equations with a smaller number of independent

variables) is studied with respect to the following algebras: AP (1, n) = Pµ , Jµ?

?

when F (u) is arbitrary; AP (1, n) = Pµ , Jµ? , D when F (u) = ?up with p an arbitrary

constant; AC(1, 3) = Pµ , Jµ? , D, Kµ when F (u) = ?u3 .

The linear equation (1), unlike the nonlinear one (1a), admits a new symmetry

operator: I = u?u so that (1) is invariant under the algebras Pµ , Jµ? , I for m = 0

and Pµ , Jµ? , I, D, Kµ for m = 0. However, until now, reductions of (1) have been

based only on subalgebras of Pµ , Jµ? and Pµ , Jµ? , D, Kµ . In this paper we take

the subalgebra Pµ , I in both cases, it allows us to reduce the hyperbolic equation (1)

to the parabolic heat equation and, in this way, we are then able to exploit the

exact solutions of the heat equation to construct solutions of the wave equation. This

is the central result of our paper. It may at first sight seem rather strange that a

Poincar?-invariant equation is reducible (with an appropriate ansatz) to one that is

e

Galilei-invariant. However, it is known (see [5]) that the Galilei algebra can be found

within the Poincar? algebra, so that one may even expect the original equation to

e

‘contain’ a Galilei-invariant one.

2. Reduction to the heat equation

In this paper we limit ourselves to (1 + 3)-dimensional time-space R(1, 3), but the

generalization of our result to higher dimensions is obvious as the reduction remains

the same.

We now turn to the construction of the ansatz which reduces (1) to the heat

equation. Equation (1) is invariant under the operators Pµ , I and is therefore also

invariant under any constant linear combination of them:

? µ ?µ + ku?u ,

where k, ? µ are constants. This latter operator then gives us the following invariant-

surface condition

? µ uµ = ku

which gives the Lagrangian system

dxµ du

=

?µ ku

and it is not difficult to show that this, in turn, is equivalent to the Lagrangian system

d(cx) du

(4)

=

c? ku

for any constant four-vector c, with cx = cµ xµ , c? = cµ ? µ . Choose now ? so that

? 2 = ? µ ?µ = 0, namely ? is light-like, and choose four-vectors ?, ?, so that

m2

? 2 = ? 2 = ?1, =?

2

(5)

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

k2

On choosing c in (4) to be ? , ?, ?, we obtain the system

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

(6)

= = = = .

0 0 0 1 ku

New solutions of the wave equation 313

The general integral of (6) is given by

u = ek( x)

(7)

v(? x, ?x, ?x),

where v is a smooth function of its arguments (we assume that all our operations

are smooth, at least locally). Treating (7) as an ansatz for equation (1), we find,

on substituting (7) into (1), writing t = ? x, y1 = ?x, y2 = ?x, performing some

elementary computations and using (5), that v satisfies the linear heat equation (we

have chosen k = 1 for convenience)

2

?2v ?2v

?v

(8)

= + 2.

2

?t ?y1 ?y2

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

which are singular for t = 0. This then leads to a similar problem for the wave

equation when ? x = 0, which is a characteristic (? 2 = 0), so that the initial-value

problem for (8) at t = 0 is related to the initial-value problem of (1) on a characteristic.

This latter is known as Goursat’s problem, and has been studied in [12], to which we

refer the reader for more details.

The linear heat equation in (1 + 1) spacetime dimensions has been studied extensi-

vely: its symmetry properties [2, 6, 7] and its conditional symmetries (also known

as non-classical symmetries [6], Q-conditional symmetries in [2]) are known. The

symmetry algebra of the linear heat equation in 1 + 2 timespace can be found in [7]

but for the sake of completeness, we give it in the following proposition.

Proposition 2. The maximal Lie point symmetry algebra of equation (8) is the

extended Galilei algebra AG3 (1, 2) with a basis given by the following vector fields

1 1

Pa = ??ya , Ga = t?ya ? ya v?v , M = ? v?v ,

T = ?t

2 2

J12 = y1 ?y2 ? y2 ?y1 , D = 2t?t + y1 ?y1 + y2 ?y2 ? v?v , (9)

12

S = t2 ?t + ty1 ?y1 + ty2 ?y2 ? t + 2

y1 + y2 v?v .

4

Remark 1. We have not included the symmetry v > v + v1 , where v1 is an arbitrary

solution of (8).

If we had considered equation (1) in R(1, 4), then we would have obtained the

linear heat equation in 1 + 3 dimensions with our reduction. Note also that there is

a Lie-algebraic reduction of (1) in R(1, 4) to equation (1) in R(1, 3), which amounts

to omitting dependency on one of the spatial variables. In this way, we are able to

use the wave equation in R(1, 4) as a bridge in constructing solutions of the wave

equation in R(1, 3) from those of the heat equation in 1 + 3 dimensions.

The invariance of equation (8) under the group G2 (1, 2) which the above algebra

generates then allows us to obtain a nine-parameter family of exact solutions whenever

one solution is given.

The commutation relations of the algebra (9) are

[Pa , Gb ] = ?ab M, [P1 , J12 ] = P2 , [P2 , J12 ] = ?P1 ,

[Pa , D] = Pa , [Pa , S] = Ga , [Pa , T ] = 0, [M, X] = 0 for all X ? AG3 (2),

[Ga , Gb ] = 0, [D, Ga ] = Ga , [T, Ga ] = Pa , [S, Ga ] = 0,

[J12 , T ] = [J12 , D] = [J12 , S] = 0, [T, D] = 2T, [T, S] = D, [D, S] = 2S.

314 P. Basarab-Horwath, L. Barannyk, W.I. Fushchych

Clearly, we see that the subalgebra Pa , Ga , M , a = 1, 2 is an ideal (maximal and

solvable, and therefore the radical of the algebra [8, 9]). Our algebra is seen to be the

semi-direct sum J12 , S, T, D + Pa , Ga , M . In turn, we can verify that S, T, D is

a semi-simple Lie algebra which we can take as being a realization of ASL(2, R), the

Lie algebra of SL(2, R). To see this, we take X1 = 1 D, X2 = 1 (T ?S), X3 = 1 (T +S)

2 2 2

as a new basis, and obtain the commutation relations of SL(2, R):

[X1 , X2 ] = ?X3 , [X2 , X3 ] = X1 , [X3 , X1 ] = X2 .

Thus we obtain

J12 , S, T, D = J12 ? S, T, D = J12 ? ASL(2, R)

which is the Lie algebra of O(2) ? SL(2, R).

The elements of the group G2 (1, 2) are considered as transformations of a space

with local coordinates (t, y1 , y2 , v) and points with these coordinates are mapped to

points (t , y1 , y2 , v ). The finite transformations defining this action are obtained by

solving the corresponding Lie equations. For the subalgebra J12 , S, T, D = J12 , X1 ,

X2 , X3 we solve the Lie equations as follows:

dt dy1 dy2 dv

= ?y2 ,

J12 : = 0, = y1 , = 0,

d? d? d? d?

t |?=0 = t, ya |?=0 = ya , v |?=0 = v

which gives the finite transformations

y1 = y1 cos ? ? y2 sin ?,

t = t, y2 = y1 sin ? + y2 cos ?, v = v.

Then we have the corresponding equations for X1 , X2 , X3

e?1 /2 t + 0 ya

X1 : t = e?1 t = , ya = e?1 /2 ya = ,

0 · t + e??1 /2 0 · t + e??1 /2

= e??1 /2 v,

v

t cosh ?2 + sinh ?2 ya

X2 : t = , ya = ,

t sinh ?2 + cosh ?2 t sinh ?2 + cosh ?2

2 2

(y1 + y2 ) sinh ?2

v = v(t sinh ?2 + cosh ?2 ) exp ,

4(t sinh ?2 + cosh ?2 )

t cos ?3 + sin ?3 ya

X3 : t = , ya = ,

cos ?3 ? t sin ?3 cos ?3 ? t sin ?3

(y 2 +y 2 ) sin ?

v = v(cos ?3 ? t sin ?3 ) exp ? 4(cos ?32 sin ?3 ) .

3

1

?t

Thus, we see that the action of the group generated by J12 , S, T, D can be given

in the form

y1 ? cos ? ? y2 ? sin ?

?t + ? y1 sin ? + y2 cos ?

t= , y1 = , y2 = ,

?t + ? ?t + ? ?t + ?

2 2

?(y1 + y2 )

v = (?t + ?)v exp

4(?t + ?)

with ?? ? ?? = 1, and ? = ±1 corresponds to the possibility of space reflections

under which (8) is manifestly invariant (the group O(2) has two components). The

parameters ?, ?, ?, ? correspond to the action of SL(2, R).

New solutions of the wave equation 315

Solving the Lie equations defined by each of the other infinitesimal generators

in (9), we obtain finite transformations such that (t, y1 , y2 , v) > (t , y1 , y2 , v ) as

follows:

Gi : t = t, yi = µi t + yi , yj = yj for j = i,

1 µ2

v = v exp ? i

t + µi yi ,

22

Pi : t = t, yi = yi ? ?i , yj = yj for j = i, v = v,

1

M : t = t, yi = yi , v = v exp ? ? .

2

3. Subalgebras and ansatzes

Having obtained and discussed the symmetry algebra of equation (8), we now pass

to listing the subalgebras of AG2 (1, 2) which are inequivalent up to conjugation by

G2 (1, 2), and giving the corresponding reduced equations. In those cases where it

is possible, we integrate these equations. The method of obtaining subalgebras up to

conjugation is described in [4, 10]; here we simply present our results. The reductions

we have obtained have been verified with MAPLE.

3.1. Reduction to ordinary differential equations by two-dimensional subal-

gebras. Here we list the subalgebras, with restrictions on any parameters entering

into the algebra, and then we give the corresponding ansatz and finally the differenti-

al equation which arises, with its solution. In all the cases, we can take the real

and imaginary parts of the solutions, as the reduced equations are linear. This is

understood when complex arguments appear.

3.1.1.

1

P2 , T + ?M (? = 0, ±1) : v = e??t/2 ?(?), ? = y1 , ? + ?? = 0.

?

2

Integrating this reduced equation, we find the following cases

for ? = 0,

? = C1 ? + C2

? ?

v + C2 exp ? v for ? = ?1,

? = C1 exp

2 2

?

v + C2 for ? = 1.

? = C1 cos

2

From these we obtain the following exact solutions of (8):

for ? = 0,

v = C1 y2 + C2

y1 y2

v = et/2 C1 exp v + C2 exp ? v for ? = ?1,

2 2

y1

v = e?t/2 C1 cos v + C2 for ? = 1

2

with C1 , C2 being arbitrary constants.

3.1.2.

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