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?2 ) + ?3 ? + ?4 and what is more b = ??2 ?1 (without loss of generality we may put
b = ?2 = 0).
In the above formulae ?1 , ?2 , ?3 , ?4 are arbitrary real constants, ?1 = 0.
Substitution of the expression for F
(12)
F = ?1 ? ln ? + ?1 ? + ?3
into Eq. 3 from the system (10) yields
?ixµ ?ixµ = Qi (?i ), i = 1, 2,
Ci = ?1 Q?1 (?i )?i ln ?i , i = 1, 2,
i
2a = ?1 a ln a + ?2 a, ?3 = 0.
Since in Eq. (12) ?1 = 0, we can rescale the function ? > k? in such a way that
F (?) takes the form F = ?1 ? ln ?. The theorem is proved.
Note. A classical example of the anti-reduction of mathematical physics equations is
the procedure of separation of variables. But the method of separation of variables can
Anti-reduction of the nonlinear wave equation 119

be effectively applied to linear second-order PDEs only, whereas the anti-reduction
procedure is evidently applicable to nonlinear differential equations.
Thus each solution of the system (7) after being substituted into ansatz (6) reduces
the nonlinear PDE (5) to two second-order QDEs

Qi (?i )?i = ??i ln ?i ,
? i = 1, 2.

Let us write down some particular solutions of Eqs. (7) under a = 1.
= ln(x2 ? x2 ), ?2 = ln(x2 + x2 );
1. ?1 0 3 1 2
= ln(x0 ? x3 ), ?2 = x1 ;
2 2
2. ?1
= x0 , ?2 = ln(x2 + x2 );
3. ?1 1 2
2 2
4. ?1 = ln(x1 + x2 ), ?2 = x3 ;
5. ?1 = x0 , ?2 = x1 ;
?1 = (x2 ? x2 ? x2 )?1/2 ,
6. ? = x3 ;
0 1 2
?1 = x0 , ?2 = (x2 + x2 + x2 )?1/2 ;
7. 1 2 3
?1 = x1 cos ?1 + x2 sin ?1 + ?2 , ?2 = x1 sin ?1 ? x2 cos ?1 + ?3 .
8.
In the above formulae ?1 , ?2 , ?3 are arbitrary smooth functions on x0 + x3 .
Let us emphasize that the above ansatzes can not be obtained within the frame-
work of the classical Lie approach (see, e.g. [5, 6]), because the maximal symmetry
group admitted by Eq. (5) is the Poincar? group P (1, 3) [2] and the general form of
e
Poincar?-invariant ansatz is given by the formula (2).
e

1. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук, думка, 1989, 336 с.
2. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., Subgroups of the Poincar? group and their
e
invariants, J. Math. Phys., 1976, 17, 977–985.
3. Fushchych W.I., Zhdanov R.Z., Yegorchenko I.A., On the reduction of the nonlinear multi-dimen-
sional wave equations and compatibility of the d’Alembert–Hamilton system, J. Math. Anal. Appl.,
1991, 161, 352–360.
4. Фущич В.И., Условная симметрия уравнений нелинейной математической физики, Укр. мат.
журн., 1991, 43, № 11, 1456–1470.
5. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
6. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986, 497 p.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 120–125.

On the new approach to variable separation
in the wave equation with potential
W.I. FUSHCHYCH, R.Z. ZHDANOV, I.V. REVENKO
Пропонується конструктивний пiдхiд до розв’язання проблем роздiлення змiнних
для двомiрного хвильового рiвняння utt ? uxx = V (x)u. У рамках цього пiдходу
описанi yci потенцiали, що допускають, роздiлення змiнних i вказанi вiдповiднi
системи координат.

A problem of variable separation (VS) in the wave equation
ux0 x0 ? ux1 x1 + V (x1 )u = 0 (1)
as considered in [1–3], consists of two problems. The first one is to describe all
functions V (x1 ) providing VS in (1) in, at least, two inequivalent coordinate systems.
The second one is to describe all coordinate-systems such that equation (1) admits
VS for a given potential V (x1 ). Surprisingly enough, the both problems are not
completely solved yet.
Our approach to the problem of VS in the wave equation (1) is based on the idea
of its reduction to two ordinary differential equations
(2)
?i = Ai (?i , ?)?i + Bi (?i , ?)?i ,
? i = 1, 2
with the use of ansatz of special structure [4–6]
(3)
u = A(x0 , x1 )?1 (?1 (x0 , x1 ))?2 (?2 (x0 , x1 )).
In the formulas (2), (3) A1 , A2 , B1 , B2 , A, ?1 , ?2 are sufficiently smooth real
functions, ? ? R1 is some parameter, no summation over i is carried out.
The formulas of the form (3) can be found in the classical works Euler, d’Alembert,
Batemen and by some other contemporary mathematicians (see, for example, the
review by Koornwinder [7]).
Definition. We say, that equation (1) admits VS in the coordinates ?1 , ?2 if substi-
tution of the ansatz (3) into (1) with subsequent exclusion of the second derivatives,
?1 , ?2 according to formulas (2) turns it into zero identically with respect to the
??
variables ?1 , ?2 , ?1 , ?2 , ?.
??
Substituting ansatz (3) into differential equation (1), expressing functions ?i in
?
terms of ?i , ?i , i = 1, 2 and splitting the obtained expression with respect to the
?
independent variables ?1 ?2 , ?1 ?2 , ?1 ?2 , ?1 ?2 we get the following system of nonli-
?? ? ?
near partial differential equations:
1) A2?1 + 2Axµ ?1xµ + AA1 ?1xµ ?1xµ = 0,
2) A2?2 + 2Axµ ?2xµ + AA2 ?2xµ ?2xµ = 0,
(4)
2A + A(B1 ?1xµ ?1xµ + B2 ?2xµ ?2xµ ) + AV (x1 ) = 0,
3)
4) ?1xµ ?2xµ = 0.
Доповiдi АН України, 1993, № 1, С. 27–32.
New approach to variable separation in the wave equation with potential 121

Hereafter, the summation over the repeated Greek indices is under-stood in the
Minkovski space M (1, 1) with a metric tensor gµ? = diag (1, ?1).
Thus to describe all potentials V (x1 ) and coordinate systems ?1 , ?2 providing VS
in (1) one has to solve nonlinear system (4). At first glance such an approach seems
to have poor prospects: to solve linear equation (1) it is necessary to integrate rather
complicated system of nonlinear partial differential equations (4). But system (4) is
overdetermined one. This fact has enabled us to construct its general solution in
explicit form. Let us emphasize that the same is true when reducing nonlinear wave
equation to the ordinary differential equation [5, 6] .
It is not difficult to show that from the forth equation of system (4) it follows that
(5)
(?1xµ ?1xµ )(?2xµ ?2xµ ) = 0.
Differentiating equations 1), 2) from (4) and using (5) we have
A1? = A2? = 0.
Consequently, the relation B1? B2? = 0 holds. Differentiating equation (3) with
respect to ?, we get
B1? ?1xµ ?1xµ + B2? ?2xµ ?2xµ = 0
or
?2xµ ?2xµ
B1?
=? .
B2? ?1xµ ?1xµ
Differentiating the above equality with respect to ?, we obtain
B1?? B2??
(6)
= .
B1? B2?
Since function Bi depends on the variable ?i and the functions ?1 , ?2 are inde-
pendent, it follows from (6) that
Bi?? = ?(?)Bi? , i = 1, 2.
Integration of the above ordinary differential equations yields
Bi = ?(?)fi (?i ) + gi (?i ), i = 1, 2.
After redefining the parameter ?, we have
(7)
Bi = ?fi (?i ) + gi (?i ), i = 1, 2.
Substituting (7) into equation (3) and splitting the obtained equality with respect
to ?, we come to the following partial differential equations:
2A + A(g1 ?1xµ ?1xµ + g2 ?2xµ ?2xµ ) + V (x1 )A = 0,
3a)
(8)
3b) f1 ?1xµ ?1xµ + f2 ?2xµ ?2xµ = 0.
Before integrating overdetermined system of nonlinear equations (4), (8), make an
important remark. It is evident, that the ansatz structure does not change with the
transformation of the form
A > Ah1 (?1 )h2 (?2 ), ?i > ?1 (?i ), (9)
i = 1, 2.
122 W.I. Fushchych, R.Z. Zhdanov, I.V. Revenko

That is why, solutions of system (4), (8) connected by relations (9) are considered
as equivalent.
Making the change (9) in equations 1), 2), 3b) by the appropriate, choice of
functions hi , ?i one can obtain f1 = f2 = 1, A1 = A2 = 0. Consequently, functions
?1 , ?2 satisfy the equations
?1xµ ?2xµ = 0, ?1xµ ?1xµ + ?2xµ ?2xµ = 0.
whence
(?1 ± ?2 )xµ (?1 ± ?2 )xµ = 0.
Integrating the above equations we get
?2 = f (?) ? g(?), (10)
?1 = f (?) + g(?),
where f , g are arbitrary smooth functions, ? = 1 (x1 + x0 ), ? = 1 (x1 ? x0 ).
2 2
Substitution of the formulas (10) into equations 1), 2) from (4) yields the following
equations for a function A(x0 , x1 )
(ln A)x0 = 0, (ln A)x1 = 0,
whence A = 1.
At last, substituting the obtained results into the equation 3b) from (8) we come to
a conclusion that the problem of integration of system (4), (8) is reduced to solution
of the functional-differential equation
df dg
V (x1 ) = [g1 (f + g) ? g2 (f ? g)] (11)
.
d? d?
And what is more, solution with separated variables (3) reads
u = ?1 (f (?) + g(?))?2 (f (?) ? g(?)). (12)
To integrate (11) it is convenient to make the hodograph transformation
(13)
? = P (f ), ? = R(g)
? ?
with P ? 0, R ? 0, equation (11) taking the form
? ? ?
g1 (f + g) ? g2 (f ? g) = P (f )R(g)R(g)V (P + R). (14)
Evidently, equality (14) is equivalent to the following relation:
? ?
(?f ? ?g )[P (f )R(g)V (P + R)] = 0
2 2

or
... ...
? ? ?? ? ? ? ?? ? ??
(P R ? R P )V + 3P R(P ? R)V + P R(P 2 ? R2 )V = 0. (15)
Without going into details of integration of equation (15) we give the final results.
Theorem 1. The general solution of (15) is given by one of the following formulas:
? ?
is an arbitrary function, R = P = ?;
1. V = V (x1 )
?
V = m(x1 + C)p2 = ?P + ?, R2 = ?R + ?;
2. ?
New approach to variable separation in the wave equation with potential 123

V = m(x1 + C)?2 , P = F (f ), R = G(g),
3.
?
F 2 = ?F 4 + ?F 3 + ?F 2 + ?F + ?, (16)
?
G2 = ?G4 ? ?G3 + ?G2 ? ?G + ?;

V = m sin?2 (x1 + C),
4. P = arctg F (f ), R = arctg G(g),

where F , G are determined by (16);

V = m sh?2 (x1 + C),
5. P = arth F (f ), R = arth G(g),

where F , G are determined by (16);

V = m ch?2 (x1 + C),
6. P = arctg F (f ), R = arth G(g),

where F , G are determined by (16);
? ?
V = m exp(??x1 ), P 2 = ?e2P + ?eP + ?, R2 = ?e2R + ?eR + ?;
7.
V = cos?2 (x1 + C)[m1 + m2 sin(x1 + C)],
8.
? ?
P 2 = ?2 sin 2P + ? 2 , R2 = ?2 sin 2R + ? 2 ;
V = ch?2 (x1 + C)[m1 + m2 sh(x1 + C)],
9.
? ?
P 2 = ? sh 2P + ? ch 2P ? ? 2 , R2 = ? sh 2R ? ? ch 2R ? ? 2 ;
V = sh?2 (x1 + C)[m1 + m2 ch(x1 + C)],
10.
? ?
P 2 = ? sh 2P + ? ch 2P ? ? 2 , R2 = ?? sh 2R + ? ch 2R ? ? 2 ;
? ? ? ?
V = m1 exp C1 x1 + m2 exp 2Cx1 , P = ?P 2 + ?, R = ?R2 + ?;
11.
V = m1 + m2 (x1 + C)?2 , P 2 = ?P 2 + ?P + ?, R2 = ?R2 ? ?R + ?;
? ?
12.
? ?
V = m, P 2 = ?P 2 + ?1 P + ?1 , R2 = ?R2 + ?2 R + ?2 .
13.

Here ?, ?i , ?i , ?, ?, m, m1 , m2 , C are arbitrary real constants.
Thus, Theorem 1 gives the complete solution of the problem of VS in wave equa-
tion (1).
Note 1. Equation (1) with potentials V = m sin?2 x1 , V = m ch2 x1 , V = m sh?2 x1
is reduced to equation (1) with the potential V = mx?2 by the changes of variables
1

1 1
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