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mentioned result gives the exhaustive description of Poincar?-invariant equations (1)
e
in the Minkowski space R(1, n).
It would be of interest to apply the technique developed in [15] to construct PDE
of the order higher than 1 which are invariant under the Poincar? algebra AP (2, 2)
e
with the basis elements (19).
In the present papers we have studied representations of the Poincar? algebra in
e
spaces with one dependent variable. But no less important is to investigate nonlinear
266 W.I. Fushchych, R.Z. Zhdanov, V.I. Lahno

representations of the Poincar? algebra in spaces with more number of dependent
e
variables [17]. Linear representations of such a kind are realized on sets of solutions
of the complex d’Alembert, of Maxwell, and of Dirac equations. If nonlinear rep-
resentations in question would be obtained, one could construct principially new
Poincar?-invariant mathematical models for describing real physical processes.
e
We intend to study the above mentioned problems in our future publications.
Besides that, we will construct nonlinear representations of the Galilei group G(1, n),
which plays in Galilean relativistic quantum mechanics the same role as the Poincar? e
group in relativistic field theory.

1. Ovsiannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982.
2. Olver P., Applications of Lie group to differential equations, Berlin, Springer, 1986.
3. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations of
nonlinear mathematical physics, Dordrecht, Kluwer Academic Publ., 1993.
4. Wigner E., Unitary representations of the inhomogenerous Lorentz group, Ann. Math., 1939, 40,
149–204.
5. Gelfand I.M., Minlos A., Shapiro Z.Ya., Representations of the rotation and Lorentz groups,
Moscow, Fizmatgiz, 1958.
6. Fushchych W.I., Nikitin A.G., Symmetry of equations of quantum mechanics, Moscow, Nauka,
1990; New York, Allerton Press, 1994.
7. Rideau G., An analytic nonlinear representation of the Poincar? group, Lett. Math. Phys., 1985, 9,
e
337–352.
8. Bertrand J., Rideau G., An analytic nonlinear representation of the Poincar? group: II. The case of
e
helicities ±1/2, Lett. Math. Phys., 1985, 10, 325–331.
9. Rideau G., Winternitz P., Nonlinear equations invariant under the Poincar?, similitude and conformal
e
groups in two-dimensional space-time, J. Math. Phys., 1990, 31, 1095–1105.
10. Yehorchenko I.A., Nonlinear representation of the Poincar? algebra and invariant equations, in
e
Symmetry Analysis of Equations of Mathematical Physics, Kyiv, Institute of Mathematics, 1992,
62–66.
11. Fushchych W.I., Zhdanov R.Z., Lahno V.I., On nonlinear representation of the conformal algebra
AC(2, 2), Dokl. AN Ukrainy, 1993, 9, 44–47.
12. Fushchych W.I., Barannik L.F., Barannik A.F., Subgroup analysis of the Galilei and Poincar? groups
e
and reduction of nonlinear equations, Kyiv, Naukova Dumka, 1991.
13. Fushchych W.I., Zhdanov R.Z., Conditional symmetry and reduction of partial differential equations,
Ukr. Math. J., 1992, 44, 970–982.
14. Fushchych W.I., Shtelen W.M., The symmetry and some exact solutions of the relativistic eikonal
equation, Lett. Nuovo Cim., 1982, 34, 498–502.
15. Fushchych W.I., Zhdanov R.Z., Yehorchenko 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.
16. Fushchych W.I., Yehorchenko I.A., Second-order differential invariants of the rotation group O(n)
and of its extensions: E(n), P (1, n), Acta Appl. Math., 1992, 28, 69–92.
17. Fushchych W.I., Tsyfra I.M., Boyko V.M., Nonlinear representations for Poincar? and Galilei
e
algebras and nonlinear equations for electromagnetic fields, J. Nonlinear Math. Phys., 1994, 1,
2, 210–221.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 267–272.

Nonlocal ansatzes
for nonlinear wave equation
W.I. FUSHCHYCH, I.M. TSYFRA
i , iii i i
i i. , i
, i i.

1. In the present paper we suggest a nonlocal ansatz
?u
(1)
= aµ? (x, u)?? (?) + hµ (x, u), µ, ? = 0, 1, 2, 3,
?xµ
for reduction of the second order nonlinear differential equation
?2u ?u
(2)
gµ? (x, u) µ ? + F x, u, =0
?xµ
?x ?x
to the system of equations for some functions ?? (?), ? = (?1 , ?2 , ?3 ). The functions
aµ? (x, u), hµ (x, u) are determined from the condition that the equation (2) is reduced
to the system of equations for ?? (?) (for more detail about the reduction method see
[1, 2]).
To illustrate the efficiency of the ansatz (1) we consider two nonlinear two-dimen-
sional equations of type (2)

u12 = u1 F1 (u1 ? u), (3)

(4)
u00 = F2 (u11 ),
2
where uµ? ? ?xµ ?x? , uµ = ?xµ , F1 , F2 are smooth functions.
?u ?u

2. For equation (3) we shall search for ansatz (1) in the form
?u ?u
(5)
= ?1 (?) + h1 (x, u), = ?2 (?) + h2 (x, u),
?x1 ?x2
h1 , h2 , ? has to be determined in the way that functions ?1 , ?2 satisfy the system of
the ordinary differential equations with a new independent variable ? [1, 2]. Substitu-
ting (5) into (3) and using the compatibility condition u12 ? u21 , we obtain
?h1 ?h1 ??1 ??
? = (?1 + h1 )[?1 + h1 ? u],
(?2 + h2 ) +
?x2 ?u ?? ?x2
?h2 ?h2 ??2 ??
= (?1 + h1 )[?1 + h1 ? u], (6)
+ (?1 + h1 ) +
?x1 ?u ?? ?x1
?h1 ?h1
h2 = h1 F1 [h1 + ?1 ? u],
+
?x2 ?u
ii , 1994, 10, P. 34–39.
268 W.I. Fushchych, I.M. Tsyfra

?h1 ??1 ??
= ?1 F1 [h1 + ?1 ? u] = R1 (?),
?2 +
?u ?? ?x2
?h2 ?h2
h1 = h1 F1 [h1 + ?1 ? u], (7)
+
?x1 ?u
?h2 ??2 ??
= ?1 F1 [h1 + ?1 ? u] = R2 (?),
?1 +
?u ?? ?x1

where R1 (?), R2 (?) are unknown functions. System (7) is a condition on functions
h1 , h2 , ?(x1 , x2 ) guaranteeing that system (6) depends on the ? only, i.e., ansatz (5)
reduces partial differential equation to a system of ordinary differential equations for
functions ?1 and ?2 . Hence, in order to describe ansatzes of type (5) it is necessary
to solve nonlinear system (7). Here, we get a particular solution of system (7) only,
namely

(8)
h1 = u, h2 = F1 [?1 (x2 )]u, ? = x2 .

It is easy to verify that solution (8) satisfies system (7) and in this case reduced
system (6) takes the form

??1
(9)
?2 (x2 ) + = ?1 (x2 )F1 [?1 (x2 )].
?x2
Having integrated the system

?u ?u ??1
= F1 [?1 (x2 )]u + ?1 (x2 )F1 [?1 (x2 )] ? (10)
= u + ?1 (x2 ),
?x1 ?x2 ?x2
one can obtain particular solutions of equation (3). The solution of equation (10) is
given by the formula

u = ??1 (x2 ) + cex1 + F1 (?1 (x2 ))dx2
(11)
,

where ?1 (x2 ) is an arbitrary smooth function and C is an arbitrary constant.
3. Now we suggest the method of construction of ansatzes (1), based on a nonlocal
symmetry of the equation

?2u ?u ?u
(12)
=F u, , .
?x1 ?x2 ?x1 ?x2

We consider the first order system

V21 + V31 V 2 = F (x3 , V 1 , V 2 ), (13)

V21 + V32 V 1 = F (x3 , V 1 , V 2 ), (14)
k
corresponding to the equation (12), where Vik ? ?V i , x3 ? u, ?xi ? V i .
?u
?x
The problem of construction of all ansatzes from the class (1) for equation (12) is
equivalent to the problem of finding all operators of the Q-conditional symmetry [1,
2, 5].
Nonlocal ansatzes for nonlinear wave equation 269

Theorem 1. The system (13), (14) is Q-conditionally invariant under the operators
Q1 = ?x1 , Q2 = ?x3 + ? 1 ?V1 + ? 2 ?V2 if and only if the functions ? 1 , ? 2 satisfy the
following equation
F
1 1 1 2 2
?V2 = 0, ?x2 = ?x1 = ?x1 = 0, ?V 1 = ,
V1 (15)
F F
? ? ? 1 1 ? ?x3 V 2 .
1 1 1 1
?x2 ?V 1 F = Fx3 + ? FV1 + 1 FV2
V V
The correctness of Theorem 1 is easily verified with the help of the infinitesimal
criterion of the Q-conditional invariance [1, 5]. Thus, arbitrary setting ? 1 (x3 , V 1 )
as a function on x3 , V 1 we get classes of nonlinearities F (x3 , V 1 , V 2 ) with which
equation (12) admits operators {Q1 , Q2 }. In the case of equation (3) ? 1 , ? 2 are as
follows: ? 1 = 1, ? 2 = F1 (V 1 ? x3 ), F = V 1 F1 (V 1 ? x3 ). It should be noted that Q2
is not a prolongation of Lie operator, but it is the nonlocal symmetry operator of the
equation (12). Operators {Q1 , Q2 } lead to the ansatz (10).
Then we consider the equation
(16)
u00 = F2 (u11 ),
where F2 is an arbitrary smooth function. Using the invariance of equation (16) under
the operators ?x0 , ?x1 , ?u , x1 ?u , x2 ?u we write it in the form of the following system
V10 = V01 , V02 = V11 , V 0 = F2 (V 2 ), (17)
where u00 ? V 0 , u01 ? V 1 , u11 ? V 2 .
Theorem 2. The system (17) is invariant with respect to the continuous group of
transformations with the infinitesimal operator
Q = ? 0 ?x0 + ? 1 ?x1 (18)
if ? 0 , ? 1 are a solution of the system of equations
?? 0 ?? 0 ?? 1 ?? 1
= = = = 0,
?x0 ?x1 ?x0 ?x1
?? 0 ?? 1 ?? 1 (19)
2
= F (V ) + ,
?V 0 2
?V 1 ?V 2
?? 1 ?? 0 ?? 0
F2 (V 2 ) +
= .
?V 1 ?V 0 ?V 2
The finite transformations
x0 = x0 + a? 0 , x1 = x1 + a? 1 (20)
correspond to the operator (18). Formulae (18), (19) give the operator of the nonlocal
symmetry of equation (16). With the help of this operator, one can construct nonlocal
ansatzes reducing the equation (16) to the system of three ordinary differential equa-
tions for three unknown functions. The analogous procedure has been called an ati-
reduction in [6].
Furthermore, the finite transformations (20) can be used for generating new solu-
tions. The transformations (20) are more general than contact ones since ? 0 , ? 1 are
the functions on u00 , u01 , u11 .
270 W.I. Fushchych, I.M. Tsyfra

For example, we shall take F2 (u11 ) = sin u11 . In this case one of the solutions of
system (19) is
c
? 0 = (V 1 )2 ? C cos V 2 + D, ? 1 = cV 1 sin V 2 + DV 1 , (21)
2
? sin x0 . Then
x0 x1
where C, D = const. We start from a solution of the equation u = 2

V 0 = sin x0 , V 1 = x1 , V 2 = x0 . (22)
Using the transformations (20) we obtain the system
c 12
(V ) ? C cos V 2 + D ,
V 0 = sin x0 + a
2
V = x1 + a[CV sin V 2 + DV 1 ],
1 1
(23)
c
V 2 = x0 + a (V 1 )2 ? C cos V 2 + D .
2
Thus, in order to find new solutions of equations (16) it is necessary to solve the
overdetermined but compatible system
c
(u01 )2 ? C cos u11 + D ,
V00 = sin x0 + a
2

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