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. 16
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4) The ansatz
2 2x0
u = exp exp(2?2 x0 ) ?(?) exp i exp ?(?) ,
? ? (43)
x0
? = (x2 )1/2 exp ? , ? = 0,
?
reduces equation (2) with ?1 = 0, ?2 = 0 to the system ODE:
?? + ?? + (n ? 1)? ?1 ?? + ??1 ? ? = 2?2 ? ln ?,
? ?? ? ?
?1
? + (n ? 1)? ? = 0, (44)
? ?
???2 ? ? ? + 2? = 0.
? ?
The systems of reduced equations (38), (40), (42), (44) are overdetermined.
Therefore it is necessary to consider their compatibility.

1. Bialynicki-Birula I., Mycielski J., Nonlinear wave mechanics, Annals of Phys. (N.Y.), 1970, 100,
1/2, 62–93.
2. Schuch D., Ghung, K.-M., Hartman H., Nonlinear Schr?dinger-type field equation for the descrip-
o
tion of dissipative systems. I. Derivation of the nonlinear field equation and one-dimensional
examples, J. Math. Phys., 1983, 24, 6, 1652–1660.
3. Nassar A., New method for the solution of the logarithmic nonlinear Schr?dinger equation via
o
stochastic mechanics, Phys. Rev. A, 1986, 33, 5, 3502–3505.
4. Fushchych V.I., Shtelen V.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear
equations of mathematical physics, Kiev, Naukova Dumka, 1989, 336 p.
5. Olver P., Applications of Lie groups to differential equations, N.Y., Springer, 1986, 497 p.
6. Chopyk V.I., Symmetry and reduction of multi-dimensional Schr?dinger equation with the loga-
o
rithmic nonlinearity, in Symmetry Analysis of Equations of Mathematical Physics, Kiev, Inst. of
Mathematics of Ukr. Acad. Sci., 1992, 54–62.
7. Collins G.B., Complex potential equations. A technique for solution, Math. Proc. Gambr. Phyl. Soc.,
1976, 80, 1, 165–171.
8. Fushchych W.I., Zhdanov R.Z., Revenko I.V., General solutions of the nonlinear wave equation and
eikonal equation, Ukr. Math. J., 1991, 43, 1471–1486 (in Russian).
9. Fushchych W., Ghopyk V., Conditional invariance of the nonlinear Schr?dinger equations, Dokl.
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Ukr. Acad. Sci., 1988, 9, 17–20 (in Ukrainian).
10. Fushchych W.I., Conditional symmetry of equations of nonlinear mathematical physics, in Symmetry
Analysis of Equations of Mathematical Physics, Kiev, Inst. of Mathematics of Ukr. Acad. Sci., 1992,
7–27.
11. Wilhelm H.F., Hydrodynamic model of quantum mechanics, Phys. Rev. D, 1970, 1, 8, 2278–2285.
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11, 1504–1509 (in Ukrainian).
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dinger equation, Ukr. Math. J., 1991, 43, 12, 1620–1628 (in Russian).
W.I. Fushchych, Scientific Works 2003, Vol. 5, 67–75.

Symmetry and exact solutions
of multidimensional nonlinear
Fokker–Planck equation
W.I. FUSHCHYCH, V.I. CHOPYK, V.P. CHERKASENKO
ii i –. -
ii i ii i
i –. i i, i
i i ’ i.

1. Let us consider equation
n n
?2
?p ? 1
=? (1)
(Ak ?) + (Bik ?) + F (?),
?t ?xk 2 ?xi ?xk
k=1 i,k=1

where ?(t, x), x = (x1 , . . . , xn ), Ak (t, x), Bik (t, x), F (?) are smooth real functions.
If F (?) = 0, (1) coincides with classical linear Fokker–Planck equation (FPE), which
finds broad application in the theory of Markov processes. In this case [1] ? is the
conditional probability density, A = (A1 , A2 , . . . An ) is a drift velocity vector, Bik are
elements of diffusion matrix B(t, x) = Bik n i,k=1 .
In the cases, when (1) (for F (?) = 0) is equivalent to the linear heat equation, it
is possible to use effectively group-theoretical analysis methods to construct solutions
of the linear FPE [2]. In other cases equation (1) for fixed Ak and Bik , as a rule, has
no nontrivial symmetry. Thus, it is impossible to apply to it symmetry methods [3].
In [4] a new interpretation for the equations like (1) was proposed, it opens
wide possibilities for application of group-theoretical methods. The idea consists in
complementing (1) with equations for coefficient functions Ak and Bik . That is we add
to (1) some system of equations for Ak and Bik , (1) turns out then to be a nonlinear
system (even if F (?) = 0). Such an extended system, as we show, can have a nontrivial
symmetry which is used to construct exact solutions of equation (1).
Therefore our paper is based on the idea of nonlinear extension of equation (1).
2. We require the components of the vector A to satisfy conditions having the
form of Euler’s equation for the ideal liquid
?Ak ?Ak
(2)
+ Al = Fk (?).
?t ?xl
?? ?F1 (?)
and Bik = Dik (D = const ? 0)
For the potential flow when Ak = , Fk = ?xk ,
?xk
equations (1) and (2) can be written as
D
?0 + ?a ?a + ??? ? (3)
?? = F (?),
2
ii , 1993, 2, . 32–42.
68 W.I. Fushchych, V.I. Chopyk, V.P. Cherkasenko

1
(4)
?0 + ?a ?a = F1 (?),
2
where ?0 = ?x0 , x0 ? t, ?a = ?xa , a = 1, n. Thus we tend to investigate symmetry
?? ??

properties and to construct families of solutions for (3), (4).
3. We assume D to be nonvanishing.
Theorem 1. Equation (3) for D > 0 is invariant under the following algebras:
1) A1 = P0 , Jab , Xa , Y, Pa , where
? ?
Jab = xa Pb ? xb Pa , {a, b} = 1, n,
P0 = , Pa = ,
?x0 ?xa
? ?
Xa = ga (x0 )Pa + ga (x0 )xa , Y = h(x0 ) ,
?? ??
(ga , h are arbitrary smooth functions) for an arbitrary F (?);
2) A2 = A1 , D , where the operator of scale transformations D has the form
2
D = 2x0 P0 + xa Pa ? I,
k
?
where I = ? ?? , for F = ??k+1 , k = 0;
3) A3 = A2 , A , where the operator of projective transformations A has the form
x2 ? 2
? nx0 I,
x2 P0 if F = ?? n +1 ,
A= + x0 xa Pa +
0
2 ??
where x 2 = x2 + x2 + · · · + x2 ;
n
1 2
4) A4 = A1 , S , where
1 1 ? n
? f (x0 )I
S = f (x0 )P0 + f (x0 )xa Pa + f (x0 )x 2
2 4 ?? 2
(f is an arbitrary smooth function), if F = 0;
5) A5 = A1 , C0 , where
C0 = exp{?x0 }I, if F = ? ln p.
Proof of this and following theorems can be made using Lie’s algorithm (see, e.g.
[5, 6]).
Remark 1. Algebra A4 coincides with A3 , if we require condition f = 0 to be
satisfied.
Remark 2. In the case D = 0 equation (1) turns into Liouville’s equation. The
question on the symmetry of (4) (if F = 0) then can be answered by the following
theorem [7].
Theorem 2. Equation (4) with D = F = 0 is invariant under infinitely-dimensional
algebra which is generated by the operator
?
X = 2f1 (x0 )P0 + f1 (x0 )xa Pa + f2 (x0 ) xa Pa + 2? +
??
x2 ? n ?
? x0 I
+ (f1 + f2 ) + f3a Pa + f3a (x0 )xa + dI + cab Jab .
2 ?? 2 ??
Symmetry and exact solutions of nonlinear Fokker–Planck equation 69

where f1 (x0 ) + f2 (x0 ) = 0, cab = ?cba , {d, cab } ? R, f1 (x0 ), f2 (x0 ), f3a (x0 ), a =
1, 4, f4 (x0 ) are an arbitrary smooth functions. Operators Xa lead to the following
finite transformations:
1
? = ? + ga (x0 )xa ? + ga (x0 )ga (x0 )?2 ,
xa = xa + ga (x0 )?, ? ?
2
dga
? = ?, x0 = x0 , xb = xb , where ga = dx0 , ? is a group parameter.
?
4. Let us now require the condition (4) on ? to be satisfied.
Theorem 3. The system of equations (3), (4) for D = 0 and arbitrary F , F1 is
invariant under the algebra
?
1) AG(1, n) = P0 , Pa , Jab , Ga , Q , where Ga = x0 Pa + xa Q, Q = ?? and additio-
nally is invariant under the following algebras:
2) AG1 (1, n) = AG(1, n), D , if F = ??k+1 , F1 = ?1 ?k , k = 0;
2 2
3) AG2 (1, n) = AG1 (1, n), A , if F = ?? n +1 , F1 = ?1 ? n ;
4) AG3 (1, n) = AG1 (1, n), B , where the operator B has the form B = I +?1 x0 Q,
if F1 = ?1 ln ?, F = 0;
5) AG4 (1, n) = AG(1, n), C , where C = exp{?x0 } ?1 Q + I , if F = ?? ln ?,
?
F1 = ?1 ln ?, ? = 0;
6) AG5 (1, n) = AG2 (1, n), I , if F = F1 = 0, where ?i are arbitrary real
constants, i = 1, 2.
Remark 3. Operator C with ?1 = 0 coincides with C0 .
Remark 4. In the case D = 0 the system (3), (4) is employed in quanturn mechanics
and is called there “the classical approximation of the Schr?dinger equations” [8]. Its
o
symmetry was investigated in [7].
5. Conditional symmetry. The system (3), (4) has conditional symmetry. The
condition which allows to enlarge symmetry of this system has the form

(5)
?? = F2 (?).

Then the system of equations (3), (4), (5) is equivalent to the following system:

?0 + ?a ?a + ??? = F (?),
1
(6)
?0 + ?a ?a = F1 (?),
2
?? = F2 (?).

Theorem 4. The system of equations (6) for arbitrary F , F1 , F2 is invariant under
the algebra AG(1, n) and additionally under the following algebras:
1) AG6 (1, n) = AG(1, n), Q1 , where Q1 = xa Pa + 2?Q if F is arbitrary and
F1 = F2 = 0;
2) AG7 (1, n) = AG(1, n), Q2 , where Q2 = x0 P0 ? ?Q for an arbitrary F2 and
F = F1 = 0;
3) AG8 (1, n) = AG(1, n), Q1 + Q2 for an arbitrary F1 and F = F2 = 0;
4) AG9 (1, n) = AG1 (1, n), Q3 , where the operator Q3 has the form: Q3 =
xa Pa + 2?Q ? k I, if F = 0, F1 = ?1 ??k , F2 = 0, k = 0;
2

5) AG10 (1, n) = AG1 (1, n), Q2 , if F = F1 = 0, F2 = ?2 ?k+1 , k = 0;
6) AG11 (1, n) = AG1 (1, n), Q1 , if F = ??k+1 , F1 = F2 = 0, k = 0;
70 W.I. Fushchych, V.I. Chopyk, V.P. Cherkasenko

7) AG12 (1, n) = AG(1, n), Q3 , if F = 0, F1 = ?1 ??k , F2 = ??k+1 , k = 0;
8) AG13 (1, n) = AG1 (1, n), Q4 , where the operator Q4 has the form: Q4 =
k+2
x0 P0 ? ?Q ? k I, if F = ?? 2 , F1 = ?1 ?k , F2 = 0, k = 0;
2

9) AG1 (1, n) if F = ??k+1 , F1 = ?1 ?k , F2 = ?2 ?k+1 , k = 0;
mk+2
10) AG14 (1, n) = AG(1, n), Q3 + mQ4 , m ? R, if F = ?? 2 , F1 = ?1 ?mk?k ,
F2 = ?2 ?k+1 , k = 0;
2+n 2+n
2
11) AG2 (1, n), if F = ?? n , F1 = ?1 ? n , F2 = ?2 ? n ;
2+n
12) AG15 (1, n) = AG2 (1, n), Q1 , if F = ?2 ? n , F1 = F2 = 0;
2+n
13) AG16 (1, n) = AG2 (1, n), Q2 , if F = F1 = 0, F2 = ?2 ? n ;
2
14) AG17 (1, n) = AG2 (1, n), Q1 + Q2 , if F1 = ?1 ? n , F = F2 = 0;
15) AG18 (1, n) = AG2 (1, n), Q1 , Q2 , if F = F1 = F2 = 0;
16) AG19 (1, n) = AG8 (1, n), B , if F = F2 = 0, F1 = ?1 ln ?;
17) AG4 (1, n) = AG(1, n), C , if F = ?? ln ?, F2 = 0;
18) AG20 (1, n) = AG6 (1, n), C0 , if F = ?? ln ?, F1 = F2 = 0, ? = 0.
Remark 5. It follows from the commutation equalities that some of above mentioned
algebras coincide (for instance, AG(1, n) and AG12 (1, n), AG7 (1, n) and AG13 (1, n)).
6. Reduction of the system (3), (4). Using the operators mentioned in Theo-
rems 3 and 4 we have constructed ansatzes and have obtained corresponding reduced
systems of equations. Some of them are adduced below (for the case of three spatial
variables, n = 3):
1) Ansatz ? = exp 2x0 ?(?1 , ?2 ), ? ? R, ? = 0,
?

x3 x2
?1 2 x0 x1 1
x0 ? 02 + ?1 = 0 ? x1 , ?2 = (x 2 ) 2 ,
?= + g(?1 , ?2 ),
2? 3? ? ?
reduces (3), (4), if F = 0, F1 = ?1 ln ? to the following system:
2 D ?1
?1
+ g11 + g22 ? + g1 ?1 + g2 ?2 ? + (?2 + ?11 + ?22 ) = 0,
? 2
2 ?g ??

<<

. 16
( 122 .)



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