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19) Ansatz ? = exp exp(?x0 ) ?(?), ? = 0,
?

2x0 x0
1
? = (x 2 ) 2 exp ?
? = exp g(?), , F = ?? ln ?, F1 = 0
? ?
reduces (6) to the system ODE:
n?1 1
? ? ?? + ?g = ?? ln ?,
?g +g
? ?
n?1
(11)
?+ ? = 0,
?
2 1 1
g ? ?g + (g )2 = 0.
? ? 2
74 W.I. Fushchych, V.I. Chopyk, V.P. Cherkasenko

Remark 6. Systems of reduced equations (7)–(11) are overdetermined.
7. Exact solutions of nonlinear Fokker–Planck equations. Below we list some
exact solutions of the FPEs in case of three spatial variables.
5
Equation (1) has a solution ? = x?3 , if Ak = xk , F (?) = ?6D? 3 ;
1) 1 x0
xk 9
3 5
? = (x2 + x2 )? 2 , Ak = , F (?) = ? D? 3 ;
2) 1 2
x0 2
x1 x2
? = x?2 , A1 = , A3 = 0, F (?) = ?3D?2 ;
3) , A2 =
1
x0 x0
xk
? = x?1 , Ak = ? 1k , F (?) = ?D?3 ;
4) 1
x0
xk
3 5
? = (x2 )? 2 , Ak = , F (?) = ?3D? 3 ;
5)
x0
x2 x2
3
? = (x2 + x2 )? 2 , A1 = , A3 = 0, F = ?2D?3 ;
6) , A2 =
1 2
x0 x0
2
D2
2 ?
c ± 2x2 {?, ?, D} = 0, ? ? R,
7) ? = exp x0 + ,
? ? D
? 2
1 1
(x 2 ) 2 xk (x 2 )? 2 , F = ?, c ? R;
Ak = ?D c ±
D2 ?
2 2 2 ?? 2
8) ? = exp x0 + y(?1 ) + z(?2 ) , Ak = , F = ?,
? D D ?xk ?
x3
?2 x0 x1
x0 ? 02 +
?= + y(?1 ) + z(?2 ),
2? 3? ?
x2 1
?1 = 0 ? x1 , ?2 = (x2 + x2 ) 2 ,
2 3
2?
2
D 2
where z = and y can be determined implicitly via relations
c3 + ?2
2? D
1 1
D2
2 2
2? 2 2? 2 D
±D ± = c ? ??1 , {?, ?} = 0;
y + ?1 + ln ? y + ?1
D ? ?? D ? ?

2 2 ??
9) ? = exp y(?1 ) + z(?2 ) , Ak = , F = 0,
D D ?xk
x3 x2 1
? = x0 x3 ? 0 + y(?1 ) + z(?2 ), ?1 = x3 ? 0 , ?2 = (x2 + x2 ) 2 ;
1 2
3 2
2x1
? (3 + 2?) ln x0 + 2c ,
10) ? = exp
x0
xk
, c ? R, F = 0;
Ak = ?? 1k +
x0
? = exp{(2x3 ? x2 )2 }, Ak = (x0 + 1)? 3k ,
11) 0
v 1
F = ?(2 2 ln 2 ? ? 4D ln ? ? 2D);
2
x3 exp(?x0 ) ? exp(?x0 )??? (x0 ) , ? ? R,
12) ? = exp
?
x1 x2
A1 = , A2 = , A3 = 0, F = ?? ln ?,
x0 x0
Symmetry and exact solutions of nonlinear Fokker–Planck equation 75

and ??? (x0 ) can be determined via relation:
exp(?x0 )
(12)
?? (x0 ) = dx0 ;
x0

x2
? exp(?x0 )??? (x0 ) ,
13) ? = exp 2 exp(?x0 ) arctg
x1
x1 x2
A1 = , A2 = , A3 = 0, F (p) = ?? ln ?,
x0 x0
where ??? (x0 ) is determined in (12);
c D
? = exp 2x1 exp(?x0 ) ? exp(?x0 )??? (x0 ) ? 2 ?
14) exp(2?x0 ) ,
?2 ?
c x2
c ? R;
A1 = exp(?x0 ), A2 = , A3 = 0, F = ?? ln ?, ? = 0;
? x0
c D
? = exp 2x1 exp(?x0 ) ? 2 exp(?x0 )??? (x0 ) ? 2 ?
15) exp(2?x0 ) ,
?2 ?
c x2 x3
A1 = exp(?x0 ), A2 = , A3 = , F = ?? ln ?; ? = 0;
?? x0 x0 ?
?2 ?
2
1
2
?
exp(?x0 ) ± 2 c + 2 ? x3 , {?, ?} = 0,
16) ? = exp
? ?? ?
D
{c, ?} ? R.
Ak = 0, F = ?? ln ?,

1. Гардинер К.В., Стохастические методы в естественных науках, М., Мир, 1986, 526 с.
2. Shtelen W.M., Stogny V.I., Symmetry properties of one- and two-dimensional Fokker–Planck equa-
tions, J. Phys. A: Math. Gen., 1989, 22, № 13, L539–L543.
3. Wolf F., Lie algebraic solutions of linear Fokker–Planck equation, Lett. Nuovo Cim., 1988, 29, № 2,
305–307.
4. Фущич В.И., О некоторых новых волновых уравнениях математической физики, в сб. Теоретико-
алгебраический анализ уравнений математической физики, Киев, Ин-т математики АН УССР,
1990, 8–12.
5. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986, 497 p.
6. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения уравнений
математической физики, Киев, Наукова думка, 1989, 336 c.
7. Чопик В.И., Симметрия одной системы уравнений квантовой механики, в сб. Симметрийный
анализ и решения уравнений математической физики, Киев, Ин-т математики АН УССР, 1988,
79–81.
8. Мессиа А., Квантовая механика, в 2-х т., М., Наука, 1978, Т.1, 470 с.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 76–79.

On nonlinear representation
of the conformal algebra AC(2, 2)
W.I. FUSHCHYCH, V.I. LAGNO, R.Z. ZHDANOV
Одержано вичерпний опис нееквiвалентних представлень алгебри Пуанкаре AP (2, 2)
та конформної алгебри AC(2, 2) у класi диференцiальних операторiв першого по-
рядку. Встановлено, що iснують лише два нееквiвалентних представлення алгебри
AP (2, 2) одне з яких є нелiнiйним. Це представлення допускає розширення до пред-
ставлення повної конформної алгебри AC(2, 2). Розглянуто деякi узагальнення.

The central problem to be solved in the framework of the classical Lie approach to
the partial differential equation (PDE) study
F (x, u, u, u, . . .) = 0 (1)
12

is the construction of its maximal symmetry group. But the inverse problem of sym-
metry analysis of PDE-description of equations invariant under given transformation
group is not of less importance. For example, relativistic field theory motion equations
have to satisfy the Lorentz–Poincar?–Einstein relativity principle. It means that consi-
e
dered equations must int under the Poincar? group P (1, 3). Consequently, to study re-
e
lativistically-invariant equations one has to study representations of the group P (1, 3)
(see e.g. [1]).
There exists vast literature on the representations of the generalized Poincar? e
groups P (n, m), n, m ? N but only a few papers are devoted to nonlinear representati-
ons [2, 3].
In the present paper we adduce results on description of unequivalent representa-
tions of the generalized Poincar? group P (2, 2) and its extention — conformal group
e
C(2, 2) acting as transformation groups in the space V = M (2, 2)?R1 , where M (2, 2)
is the Minkowski space with the metric tensor
?
? 1, ? = ? = 1, 2;
g?? = ?1, ? = ? = 3, 4;
?
0, ? = ?.
Lie algebra of the above conformal transformation group (called conformal algebra
AC(2, 2)) has the basis elements of the form
4
? ?
(2)
Q= ?a (x, u) + ?(x, u)
?xa ?u
a=1

that satisfy the following commutational relations:
[P? , P? ] = 0, [P? , J?? ] = g?? P? ? g?? P? ,
[J?? , J?? ] = g?? J?? + g?? J?? ? g?? J?? ? g?? J?? , (3)
[D, J?? ] = 0, [P? , D] = P? , [K? , J?? ] = g?? K? ? g?? K? ,
Доповiдi АН України, 1993, № 9, С. 44–47.
On nonlinear representation of the conformal algebra AC(2, 2) 77

[P? , K? ] = 2(g?? D ? J?? ), [D, K? ] = K? , [K? , K? ] = 0.

Here ?, ?, ?, ? = 1, 4.
Let us note that operators P? , J?? form generalized Poincar? algebra AP (2, 2)
e
which is a subalgebra of the conformal algebra.
Definition 1. Set of operators P? , J?? , D, K? of the form (2) satisfying the commu-
tational relations (3) is called a representation of the conformal algebra AC(2, 2).
Definition 2. Representation of the algebra AC(2, 2) is called linear if coefficients of
its basis operators (2) satisfy the conditions
(4)
?? = ?? (x), ? = a(x)u.
If conditions (4) are not satisfied, representation is called nonlinear.
It is well-known that commutational relations are not altered by the change of
variables
(5)
x? = fa (x, u), u = g(x, u).
That is why two representations {P? , J?? , D, K? } and {P? , J?? , D , K? }, are called
equivalent provided they are connected by the relations (5).
Theorem 1. There exist only two unequivalent representations of the Poincar? e
algebra AP (2, 2):
J?? = g?? x? ?? ? g?? x? ?? , (6)
1. P? = ? ? ,

P? = ?? , J12 = ?x2 ?1 + x1 ?2 + ?u ,
2.
J13 = x3 ?1 + x1 ?3 + cos u?u ,
J14 = x4 ?1 + x1 ?4 ? ? sin u?u ,
(7)
J23 = x3 ?2 + x1 ?3 + sin u?u ,
J24 = x4 ?2 + x2 ?4 + ? cos u?u ,
J34 = x4 ?3 ? x3 ?4 + ??u , ? = ±1.

Here ?? = ?/?x? , ?u = ?/?u; ?, ?, ?, ? = 1, 4, the summation over the repeated
indices from 1 to 4 is understood.
Because of the lack of the space we adduce only a sketch of the proof.
Since operators P? , ? = 1, 4 commute, there exists a change of variables (5)
reducing these to the form P? > P? , ? = 1, 4 [4]. From the commutational relations
[P? , J?? ] = g?? P? ? g?? P? it follows that operators J?? are of the form J?? =
g?? x? ?? ?g?? x? ?? +???? (u)?? +??? (u)?u , where ???? , ??? are some smooth functions,
?, ?, ? = 1, 4.
Substituting the obtained result into the third equality from (3) we get a system
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