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(1)
?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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