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where functions V and F satisfy the system:
F0 + V0 ? 4?bx0 (Fa + Va )Fa ? 2?bx0 ?F = 0,
(21)
?(Fa + Va )(Fa + Va ) + ?(?F + V ) + 2bx0 (F0 ? 2?bx0 Fa Fa ) = 0.
Case 1. The functions V and F satisfy the conditions:
(22)
F = f1 (x0 ), V = f2 (x0 ) + ?(?), ? = ?(x).
Substitution of the expression (22) into (21) yields the ODE
(? + ?2 )?1 (?) + ??2 (?) = 2b??1 ?, (23)
?? ?
where
(24)
?a ?a = ?1 (?), ?? = ?2 (?),
and
f1 = c2 ? c1 x?1 , f2 = c1 x?1 , c1 , c2 ? R. (25)
0 0

Note 4. The necessary conditions of compatibility and the general solution of system
(24) construct in papers [7, 8].
For the partially case ? = ?a xa , ?a ?a = 1, ?a ? R, a = 1, n, the equation (23)
has the form:
? + ?2 = 2b??1 ?. (26)
??
This equation by means of change of variables
?2 = ?(?)
?
62 W.I. Fushchych, V.I. Chopyk

is reduced to a linear equation:
?(?) + 2?(?) = 4b??1 ?.
?

The last equation can be easily integrated and the result is as follows:
?1/2
1
d? = (2b??1 )1/2 d?,
? + c exp(?2?) ? c ? R. (27)
2
When c = 0 we get from (27) the following solution of (26):
b 1
(? + c3 )2 + , c3 ? R. (28)
?=
2? 2
Summarizing results (20), (22), (25), (28) we write down the exact solution of equati-
on (1):
b 1
(?a xa + c3 )2 + c2 + ? 2ib(c2 x0 ? c1 ) ,
u = exp
2? 2
where ci ? R, i = 1, 2, 3, ?a ?a = 1.
Case 2. V = 0 and F satisfy the overdetermined system:
F0 ? 4?bx0 Fa Fa ? 2?bx0 ?F = 0,
?Fa Fa + ??F + 2bx0 F0 ? 4?b2 x2 Fa Fa = 0.
0

For this case the ansatz (2) has the form:
u = exp{1 ? 2ibx0 )F (x0 , x)}. (29)
Consequence. The ansatz (29) gives the solutions of the equation (1) if the real
function F satisfy:
Ft ? ?bFa Fa = 0, t = x2 . (30)
Ft + ?b?F = 0, 0

The system (30) have non-trivial symmetry properties:
Theorem 3. The overdetermined system (30) is invariant with respect to the exten-
ded Galilei algebra having basis elements:
? ?
, t = x2 , Pa , Jab , Pn+1 =
Pt = ,
0
?t ?F
= F Pa ? xa (2?b)?1 Pt , D(1) = 2t?t + xa Pa .
G(1)
a
(1)
Note 5. The operator Ga generates the transformation:
t > t = t ? (2?b)?1 ?a xa ? (4?b)?1 ?a , xb > xb = xb ,
2

xa > xa = xa + ?a F, F > F = F,
where ?a is a group parameter.
2) For reduction of the equation (2) (?1 = 0) by operator C it is necessary to
change of variables:
1
exp(?2?2 x0 )(ln |u| ? (2i?1 )?1 ?2 ln(u/u? )),
W = F (x0 , x, ?), ?=
2 (31)
1
V = ?1 ln |u| ? i ?2 ln(u/u? ).
2
Symmetry analysis and ansatzes for the Schr?dinger equations
o 63

Substituting (31) for the partially case F? = 1 into the equation (2) we get:

(2?1 )?1 V0 ? exp(2?2 x0 )(F0 ? W0 ) + 2?[(2?)?1 Va ?
? exp(2?2 x0 )(Fa ? Wa )((2?2 )?1 Va + ?1 (?2 )?1 exp(2?2 x0 )(Fa ? Wa )] +
+ (2?2 )?1 ?[?V + 2?1 exp(2?2 x0 )(?F ? ?W )] ? (?1 )?1 ?2 V = 0,
?[(2?1 )?1 Va ? exp(2?2 x0 )(Fa ? Wa )(2?1 )?1 Va ? exp(2?2 x0 )(Fa ? Wa )] +
+ ?[(2?1 )?1 ?V ? exp(2?2 x0 )(?F ? ?W )] ? (2?2 )?1 [V0 +
+ 2?1 exp(2?2 x0 )(F0 ? W0 )] ? ?(4?2 )?1 [Va + 2?1 exp(2?2 x0 )(Fa ? Wa )] ?
2
? [Va + 2?2 exp(2?2 x0 )(Fa ? Wa )] ? V = 0,

and the operator C has the form:
?
(32)
C= .
?W
From (31), (32) follows that the solutions of the equation (2) (with ?1 , ?2 = 0) we
can find in the form:

u = exp (2?1 ?2 )?1 V (?2 + i?1 ) ? (?1 )?1 F exp(2?2 x0 )(?2 ? i?1 ) ,

where the real functions V and F satisfy the system:

(2?1 )?1 V0 ? exp(2?2 x0 )F0 + 2?[(2?1 )?1 Va ? exp(2?2 x0 )(Fa ? (2?2 )?1 Va ) +
+ ?1 (?2 )?1 exp(2?2 x0 )Fa ] +
+ (2?2 )?1 ?[?V + 2?1 exp(2?2 x0 )?F ] ? (?1 )?1 ?2 V = 0,
(33)
?[(2?1 )?1 Va ? exp(2?2 x0 )Fa ] + ?[(2?1 )?1 ?V ? exp(2?2 x0 )?F ?
? (2?2 )?1 [V0 + 2?1 exp(2?2 x0 )F0 ] ?
? ?(4?2 )?1 [Va + 2?1 exp(2?2 x0 )Fa ]2 ? V = 0.
2

Case 1: V = 0. For this case the ansatz

u = exp ?(?1 )?1 F exp(2?2 x0 )(?2 ? i?1 )

reduces the equation (2) when ?1 = 0 to the system:

F0 + ??1 (?2 )?1 ?F = 0,
F0 + ??1 (?2 )?1 exp(2?2 x0 )Fa Fa = 0.

Case 2: F = 0. For this case the ansatz

u = exp (2?1 ?2 )?1 V (?2 + i?1 )

reduces the equation (2) with ?1 = 0 to the overdetermined system:

V0 + ?(?2 )?1 Va Va + ??1 (?2 )?1 ?V ? 2?2 V = 0,
(34)
V0 + ?(?2 ? ?2 )(2?1 ?2 )?1 Va Va ? ??2 (?1 )?1 ?V + 2?2 V = 0.
1 2

For the partially case ?2 = ?2 the system (34) has the form:
1 2

V0 + ?(2?2 )?1 Va Va = 0, V0 + ??V ? 2?2 V = 0,
64 W.I. Fushchych, V.I. Chopyk

and for the partially case 3?2 = ?2 this system has the form:
1 2
v
V0 + ?(2?2 )?1 Va Va ? ?2 V = 0, 3V0 ? ??V = 0.
5. Conditional symmetry. The symmetry of the equations (1), (2) can be extended
essentially, if we put a certain additional condition on its solutions (see [4, 9, 10]).
As to Schr?dinger equations with the logarithmic nonlinearity one of such additional
o
conditions is vanishing of the interior potential [11] that is equivalent to the following
condition:
|u| = (uu? )1/2 . (35)
?|u| = 0,
Theorem 4. The equation (1) is conditionally invariant with respect to the following
algebras:
1) AG5 (1, n) = AG3 (1, n), Q1 ,
where
i
ln(uu??1 )Q
Q1 = x0 P0 + xa Pa ?
2
2) AG5 (1, n) = AG(1, n), Q2 ,
where the operator Q2 is of the form [9]:
i
ln(uu??1 )Q + x0 P0
Q2 =
2
if the module of the function u satisfies the condition
??|u| = 2b|u| ln |u|. (36)
Note 6. The operator Q1 generates the following finite transformations:
u > u = |u|(uu??1 )1/2?1 ,
x0 > x0 = ?1 x0 , xa > xa = ?1 xa ,
and the operator Q2 generates the following transformations:
u > u = |u|(uu??1 )?1/2?2 ,
x0 > x0 = ?2 x0 , xa > xa = xa ,
where ?1 and ?2 are group parameters.
Theorem 5. The equation (2) is conditional invariant with respect to the algebra:
AG7 (1, n) = AG4 (1, n), Q3 ,
where
Q3 = Q1 ? Q2 = xa Pa ? i ln(uu??1 )Q,
and the operator C is of the form C = exp(2?2 x0 )I. The additional condition has
the form (34).
Note 7. The operator Q3 generates the transformations:
u > u = |u|(uu??1 )?3 ,
x0 > x0 = x0 , xa > xa = ?3 xa , a = 1, n.
Symmetry analysis and ansatzes for the Schr?dinger equations
o 65

The following theorems can be proved by means of conditional invariance algori-
thm (see e.g. [5, 10]).
So we can see that the additional conditions (34) and (35) expand the symmetry
of the equations (1), (2).
6. Applications: non-Lie reduction. In this section we consider some non-Lie
ansatzes for the equations (1), (2) which cannot, be obtained by means of classical
Lie approach. The examples of non-Lie reduction of the Schr?dinger equations with
o
degree nonlinearity are adduced in [12, 13].
1) The ansatz
u = x2 ?(?1 , ?2 ) exp{i[?a xa ? 4bx0 ln x0 + x0 ?(?1 , ?2 )]},
0
x1 x2 (37)
, ?a ? R, a = 1, n
?1 = , ?2 =
x2 x0
reduces the equation (1) to the system:
2? ? ?1 ?1 ? ?2 ?2 + 2??1 ?1 + 2??2 ?2 + ??(?11 + ?22 ) = 0,
(38)
?11 + ?22 = 0,
??2 + ??2 ? ?1 ?1 ? ?2 ?2 = 4b ? ??a ?a ? 2b ln ?, a = 3, . . . , n, ?a ? R.
1 2

2) The ansatz
x2
? 4bx0 ln x0 + x0 ?(?1 , ?2 )
1
x2 ?(?1 , ?2 ) exp (39)
u= i ,
0
4?x0
where
x2 + x2
x1 x3
? arctg , ?2 = 2 3
?1 =
x0 x2 x0
reduces the equation (1) (when n = 3) to the system:
?2 ?2
2? ? ?1 ?1 (1 + ?2 ) ? ?2 ?2 + ?2 ?2 + ??2 ?2 + ??11 (1 + ?2 ) + ??22 = 0,
?2 2
(40)
?11 (1 + ?2 ) + ?2 ?22 + ?2 ?2 = 0,
?2
+ ??2 ? ?2 ? + ? ? 4b + 2b ln ? = 0.
?2 )?2
?(1 + 1 2

3) The ansatz
x2
? 4bx0 ln x0 + x0 ?(?1 , ?2 )
1
x2 ?(?1 , ?2 ) exp
u= i ,
0
4?x0 (41)
x2 x3
?1 = , ?2 =
x0 x0
reduces the equation (1) to the system (when n = 3):
1
2? ? ?1 ?1 ? ?2 ?2 + 2??1 ?1 + 2??2 ?2 + ? + 2??(?11 + ?22 ) = 0,
2
(42)
?11 + ?22 = 0,
??2 + ??2 ? ?1 ?1 ? ?2 ?2 + ? ? 4b + 2b ln ? = 0.
1 2

Note 8. The ansatzes (37), (39), (41) are obtained as a consequence of conditional
invariance of the equation (1) respect to the algebra AG5 (1, n).
66 W.I. Fushchych, V.I. Chopyk

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