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2 2
g1 + g2 = ? ln ? + ?1 , where gi = , ?i = , i = 1, 2.
? ??i ??i
1
2x0
?(?), ? = (x 2 ) 2 , ? = 0
2) Ansatz ? = exp ?

?1 2
x + g(?), with F = 0, F1 = ?1 ln ?
?=
?0
reduces (3), (4) to the system:
n?1 n?1
2 D
+ g +g ?+g ? + ?+ ? = 0,
? ? 2 ?
dg d?
(g )2 = 2?1 ln ?, where g = , ?= .
d? d?
3) F = 0, F1 = ?1 ln ?,
2x0 x2 x3
1
?,
?1 = (x2 + x2 ) 2 ,
? = exp ?(?, ?2 ), ?2 = arctg
1 2
? x1 ?
?1 2
{?, ?} = 0,
?= x + g(?1 , ?2 ),
2? 0
Symmetry and exact solutions of nonlinear Fokker–Planck equation 71

2 ?1 ?2 ?2
+ g1 ?1 + g11 ? + g1 ?1 + g2 ?2 (?1 + ? ?2 ) + g22 ?(?1 + ? ?2 ) +
?
D ?1 ?2
+ (?11 + ?1 ?1 + ?22 (?1 + ? ?2 )) = 0,
2
?1
?2
g1 + g2 (?1 + ? ?2 ) =
2 2
ln ?.
2
4) Ansatz
2
{?, ?} = 0, ? ? R,
? = exp exp(?x0 ) ?(x3 ), ? = 2g(x3 ),
??
reduces (3), (4), if F = ?? ln ?, F1 = 0, to the following system:
2?
?+ ? ln ? = 0,
D
g = 0, D = 0.
5) Ansatz ? = exp{2x1 exp(?x0 )}?(x0 ),
x2 + x2
?1
x1 exp{?x0 } + 2 3
?= + g(x0 ), ?=0
? 2x0
reduces (3), (4), if F = ?? ln ?, F1 = ?1 ln ?, to the system ODE:
?1
? + 2x?1 ? + 2 D + exp exp{2?x0 }? = ?? ln ?,
0
?
?2
g + 12 exp{2?x0 } = ?1 ln ?.
2?
exp(?x0 ) ?(?1 , ?2 ), {?, ?} = 0, ? ? R,
2
6) Ansatz ? = exp ? x3
?1 1
exp{?x0 }x3 + g(?1 , ?2 ), ?1 = x0 , ?2 = (x2 + x2 ) 2 ,
?= 1 2
?
reduces (3), (4), if F = ?? ln ?, F1 = ?1 ln ?, to the following system:
?1 D
?1 ?1
exp{2??1 } ? + (?22 + ?2 ?2 ) = ?? ln ?,
?1 + g2 ?2 + ?2 g2 + 2
?? 2
2
?
2g1 + g2 = ?1 ln ? ? 21 2 exp{2??1 }.
2
2? ?
7) F = ?? ln ?, F1 = ?1 ln ?,
x2 1
exp(?x0 ) ?(?1 , ?2 ), ?1 = x0 , ?2 = (x2 + x2 ) 2 ,
? = exp 2 arctg 1 2
x1
x2
?1 x2
+ 3 + g(?1 , ?2 ), ? = 0,
?= exp(?x0 ) arctg
? x1 2x0
2?1 D
?2
exp{2??1 } ? + g2 ?2 +
?1 + ?2 g2 + D + ?22 +
? 2
D ?1
+ ? ?2 + g22 ? = ?? ln ?,
22
2
?1 ?1
2g1 + g2 = ?1 ln ? ? ? exp{??1 }
2
.
?2
72 W.I. Fushchych, V.I. Chopyk, V.P. Cherkasenko

8) F = ?? ln ?, F1 = ?1 ln ?,

2 ?1 1
? = (x 2 ) 2 ,
? = exp exp(?x0 ) ?(?), ?= exp(?x0 ) + g(?),
?? ??

n?1 n?1
D
g ? +? g + g + ?+ ? = ?? ln ?,
? 2 ?
(g )2 = ?1 ln ?.

x3 x2
9) Ansatz ? = ?(?), ? = x0 x3 ? 30 + g(?), ? = x3 ? 0
for arbitrary F (?), F1 (?)
2
reduces (3), (4) to the following system:

D
g ? +g ?+ ? = F (?),
2
(g )2 + 2? = F1 (?).

x2 1
+ g(?1 , ?2 ), ?1 = x0 , ?2 = (x2 + x2 ) 2 , F (?), F1 (?) are
3
10) ? = ?(?1 , ?2 ), ? = 1 2
2x0
arbitrary functions,

D
?1 ?1 ?1
(?22 + ?2 ?2 ) = F (?) ? ?1 ?,
?1 + g2 ?2 + ?(g22 + ?2 g2 ) +
2
2
g1 + g2 = F1 (?).
v 1
11) ? = ?(?), ? = g(?) ? x0 ? x2
? = (x2 + x2 ) 2 , F (?), F1 (?) are
x1 ,
2 arctg 1 2
arbitrary functions,

D
g (? + ? ?1 ?) + g ? + (? + ? ?1 ? ) = F (?),
2
(g )2 = 2F1 (?).

x2
3
12) ? = ?(x0 ), ? = + g(x0 ),
2x0

? + ? ?1 ? = F (?),
(g ) = F1 (?),

13) ? = ?(x3 ), ? = g(x3 ) ? x0 ,

D
g ? +g ?+ ? = F (?),
2
2 + (g )2 = 2F1 (?).

x2 +···+x2
x2 +···+x2
0 ? k ? 2,
14) Ansatz ? = xm ?(?), ? = 1 k+1 k+l
+g(?), where ? = ,
k
0 2x0 2x0
1
?m
m?1
1 ? l ? 3 ? k reduces (3), (4), if F = ?? , F1 = ?? , to the system ODE:
m

m?1
(k + m)? + ? (2?g ? ?) + ?(lg + 2?g ) + D(l? + 2?? ) = ?? ,
m

1
(g )2 ? g = ?1 ? ?1 ? ?m
.
Symmetry and exact solutions of nonlinear Fokker–Planck equation 73

15) Ansatz ? = x2 ?(?1 ; ?2 ), ?1 = x1 , ?2 = x2 , ? = 2?1 x0 ln x0 +?x3 +x0 g(?1 , ?2 ),
0 x0 x0
? ? R reduces system (6), if F = 0, F1 = ?1 ln ? to the following system:
?2
2?1 + g ? ?1 g1 ? ?2 g2 + = ?1 ln ?,
2
(7)
?11 + ?22 = 0,
2? ? ?1 ?1 ? ?2 ?2 + g1 ?1 + g2 ?2 + (g11 + g22 )? = 0.
16) F = 0, F1 = ?1 ln ?, F2 = 0,
x2
? = 1 + ?1 x0 ln x0 + x0 g(?1 , ?2 ),
x2 ?(?1 , ?2 ),
?= 0
2x0
x1 x3 1
?2 = x?1 (x2 + x2 ) 2 ,
? arctg ,
?1 = 2 3
0
x0 x2
12 ?1
?2
?1 + g ? ?2 g2 + g1 (1 + ?2 ) + g2 =
2
ln ?,
2 2
(8)
?2 2
?11 (1 + ?2 ) + ?22 ?2 + ?2 ?2 = 0,
?2 ?2
3? ? ?2 ?2 + ?1 (1 + ?2 )?1 + g2 ?2 + g11 (1 + ?2 )? + (g22 + ?2 g2 )? = 0.
x2
17) Ansatz ? = x2 ?(?1 , ?2 ), ?1 = x2 , ?2 = x3 , ? = 2x10 + ?1 x0 ln x0 + x0 g(?1 , ?2 )
0 x0 x0
reduces system (6), if F = 0, F1 = ?1 ln ?, to the following system:
12 ?1
?1 + g ? ?1 g1 ? ?2 g2 + (g1 + g2 ) =
2
ln ?,
2 2
(9)
?11 + ?22 = 0,
3? + g1 ?1 + g2 ?2 + (g11 + g22 )? ? ?1 ?1 ? ?2 ?2 = 0.
x2
18) Ansatz ? = exp{exp(?x0 ) ln x1 }?(?1 , ?2 ), ? = 2x20 + x2 g(?1 , ?2 ), ?1 = x0 ,
1
?2 = x2 ? xxx3 reduces system (6), if F = ?? ln ?, F1 = 0, to the system:
0
x1 1

?1
2g1 + 2?2 g2 (?1 ? 2) + g 2 + g2 (1 + ?1 + ?2 ) = 0,
2 2 2

1 1
exp(??1 )? + ?22 (1 + ?1 + ?2 ) + ?2 ?2 (1 ? 4 exp(??1 )) +
2 2
2 2
(10)
+ 2? exp(2??1 ) = 0,
?1
?1 + ?2 (?1 ? 4?2 g) + g?(2 + 4 exp(??1 )) ? 2(1 + exp(??1 ))?2 g2 ? +
?1
2 2
+ (g22 ? + g2 ?2 )(1 + ?1 + ?2 ) + ?1 ? = ?? ln ?.
2
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