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ur ur
ur u1 ur ur ?gµ?
µ µ µ? ??
= r ? 1, + (1 ? ?) r + ?
r r
?ur ur
?µ ?µ? = ,
1?n
µ? ??
ur
u u u
Sjk , Rk будуються аналогiчно (3).
AC(1, n), ? = 0:
(u1 u1 )?2k Sjk (wµ? , wµ? ), Rk (ur , wµ? )(u1 u1 )?2k+1 ,
ur , 1 r 1
?? µ ??
r = 2, . . . , m, j = 0, . . . , k, k = 1, . . . , n + 1.
r
Конформно-коварiантнi тензори wµ? мають вигляд
gµ? r
? ur (ur ur + ur ur )
wµ? = u? ur ur +
r
(5)
u
1 ? n ??
? µ? ? µ ?? ? ?µ


(по r пiдсумовування немає).
Одержанi результати дозволяють побудувати новi нелiнiйнi багатовимiрнi кон-
формно-iнварiантнi рiвняння. Наприклад, рiвняння
1
2u ? uµ u? uµ? = (u? u? )2 F (u),
u? u?
n?1
лiва частина якого є згорткою тензора wµ? (5), F — довiльна функцiя, iнварiантне
вiдносно алгебри AC(1, n), ? = 0.

1. Фущич В.И., Егорченко И.А., О симметрийных свойствах комплекснозначных нелинейных вол-
новых уравнений, Докл. АН СССР, 1988, 298, № 2, 347–351.
2. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 563–572.

The symmetry and exact solutions
of the non-linear d’Alembert equation
for complex fields
W.I. FUSHCHYCH, I.A. YEHORCHENKO
The non-linear wave equations for the complex scalar field invariant under a conformal
group are constructed and multiparametrical exact solutions of certain non-linear
complex d’Alembert equations are found.

1. The non-linear wave equation
The non-linear wave equation
pµ pµ u + F (u) = 0
for the real function u = u(x0 ? t, x1 , . . . , xn ) is invariant under the extended Poincar?
e
algebra A1 P (1, n) ? Pµ , Jµ? , D
?
Jµ? = xµ p? ? x? pµ , (1)
Pµ = pµ = igµ? ,
?x?
where D is the dilation operator (D = xµ pµ + ?upu ) iff F (u) = ?uk [4].
The classical and quantum scalar field, as is well known (see [1]), is described by
the wave equation for the complex function u. Therefore it is interesting to construct
the classes of non-linear wave equations invariant under wider groups than the Poi-
ncar? group. In the case of real fields, as was shown by Fushchych and Serov [4],
e
there exist only two classes of such non-linear fields. In the complex case there are
wide classes of fields invariant under groups which include the Poincar? group P (1, n)
e
as the subgroup.
In the present paper for the classical complex field u we construct the non-linear
second-order wave equations
?u?
?u
pµ pµ u + F (u, u? , u? , u? ) = 0, u? ?
u? ? ?, µ = 0, 1, . . . , n (2)
, ,
? ?
?x? ?x?
(the asterisk designates the complex conjugation and we indicate the sum by repeating
indices: pµ pµ = p2 ?p2 ?· · ·?p2 ) invariant under the following Lie algebras (containing
n
0 1
as subalgebra the Poincar? algebra AP (1, n) = Pµ Jµ? with the basic elements (1)):
e
(1) (1)
A1 ? A1 P (1, n) ? Pµ , Jµ? , D1 .
The dilation operator D1 has the form
? ?
D1 = xµ pµ ? ?(upu + u? pu? ), pu = ?i pu? = ?i
, .
?u?
?u
(2) (2)
A1 ? A1 P (1, n) ? Pµ , Jµ? , D2 .
J. Phys. A: Math. Gen., 1989, 22, 2643–2652.
564 W.I. Fushchych, I.A. Yehorchenko

The dilation operator D2 has the form
D2 = xµ pµ ? ?(pu + pu? ).

A2 ? A2 P (1, n) ? Pµ , Jµ? , D1 , Q .

The operator of charge has the form
Q = u? pu ? upu? .
(1)
A3 ? A(1) C(1, n) ? Pµ , Jµ? , D1 , Kµ .
(1)


(1)
The operators Kµ generating the conformal transformations have the form
Kµ = 2xµ D1 ? x? x? pµ .
(1)


(2)
A3 ? A(2) C(1, n) ? Pµ , Jµ? , D1 , Kµ , Q .
(1)


(3)
A3 ? A(3) C(1, n) ? Pµ , Jµ? , D2 , Kµ .
(2)


Kµ = 2xµ D2 ? x? x? pµ .
(2)


To describe the invariant equations of the form (2) we need the differential inva-
(1) (3)
riants of the zero and first order for the algebras A1 , . . . , A3 . As is well known
(see, e.g. [7]) these invariants are solutions of the system
1
? ?
(3)
X i ?(u, u , u? , u? ) = 0,
1
where X i are the first prolongations of the basis operators of the corresponding
algebras.
Not going into details we adduce the explicit form of the invariants for the algebras
AP (1, n) : u, u? , r1 = u? u? , r2 = u? u? , r3 = u? u? ,
? ??
2
u r1 r3 r1
(1)
A1 : , , , 2(??1) ,
u? r2 r2 u
r1 r3 ?/2
A1 : u ? u? ,
(2)
, ,r exp u,
r2 r2 1
AP (1, n) ? Q : u2 + u?2 , r1 + r3 , r2 ? r1 r3 , R = u?2 r1 ? 2uu? r2 + u2 r3 ,
2

r2 ? r1 r2
(r1 + r3 )2 2
R
A2 : , , ,
(u2 + u?2 )??1 (r1 + r3 )2 (u2 + u?2 )(r1 + r3 )
(4)
u R
(1)
A3 (? = 0) : , ,
u? u4?2/?
r1 r3
A3 (? = 0) : u, u? ,
(1)
, ,
r2 r2
A3 (? = 0) : R(u2 + u?2 )1/??2 ,
(2)

r2 ? r1 r3
2
R
?2
(2) 2
A3 (? = 0) : u + u , , ,
(r1 + r3 )2 r1 + r3
(? = 0) : u ? u? , (r1 ? 2r2 + r3 )?/2 exp u.
(3)
A3
The symmetry and exact solutions of the non-linear d’Alembert equation 565

These systems of invariants are complete when n ? 3.
The classification of the non-linear equations for the complex scalar field invariant
under the enumerated algebras gives the following theorem.
Theorem. Equation (2) is invariant under the algebras
when F = ?(?),
AP (1, n)
(1)
when F = u1?2/? ?(?),
A1
(2)
when F = exp(u)?(?),
A1
when F = (u2 + u?2 )?1/? (uf (?) + iu? g(?)),
A2
2? + n ? 1
(1)
r1 + u1?2/? ?(?),
when F =
A3 , ? = 0
2?u
when ? = 0 there are no invariant equations of the form (2);
1?n
when F = (u2 + u?2 )2/(n?1) (uf (?) + iu? g(?))
(2)
A3 , ? =
2
(when ? = (1 ? n)/2 there are no invariant equations of the form (2));
n?1 2
(3)
r1 + exp ? u ?(?)
when F =
A3 , ? = 0
2? ?
(here we designate as f and g arbitrary real and as ? arbitrary complex functions,
? are invariants of the corresponding algebras).
To prove the theorem it is necessary to use the Lie invariance condition in the
form
2
X iL =0
L=0
L? = 0

2
where L = 2u ? F (u, u? , u? , u? ) (2u = pµ pµ u), X i are the second prolongations of
?
the basis elements of the algebras being considered, which we resolve with respect to
the unknown function for every algebra.
A similar theorem can be formulated and proved for the system of two wave
equations for the pair of real functions.
The classification of the general quasilinear Poincar?-invariant equation for the
e
complex scalar function is adduced by Fushchych and Yehorchenko [5].
2. The solutions of wave equations for the complex function
Let us consider the equation
2u = F (u, u? ) (5)
which is invariant under the Poincar? algebra (1). Its solutions can be found with
e
the help of the reduction with respect to subalgebras of AP (1, n) as was done in the
real case by Fushchych and Serov [4] or Winternitz et al [8] but such reduction leads
mostly to systems of ordinary differential equations not solvable in quadratures; one of
the ways to avoid this difficulty was suggested by Grundland and Tuszynski [6]. To
find the exact solutions of (5) it is advisable to search especially for ansatze leading
to systems of differential equations solvable in quadratures.
566 W.I. Fushchych, I.A. Yehorchenko

Using the ansatz (see, e.g., [3, 4])
u? = ?? (?), (6)
u = ?(?), ? = ?(x)
we come to the system
?µ ?µ ? + 2? ? = F (?, ?? ),
? ?
(7)
d?
?µ ?µ ?? + 2? ?? = F ? (?, ?? ),
? ? ?=
?
.
d?
The condition of separation of variables in the system (7) is that the new variable
? must satisfy the conditions
2? = ?(?), (8)

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