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the Schr?dinger group, Theor. and Math. Fisika, 1983, 56, 3, 387–394.
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Ukrainian Acad. Sci., 1990, 7, 24–28.
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W.I. Fushchych, Scientific Works 2003, Vol. 5, 357–364.

?
Galilei invariant nonlinear Schrodinger type
equations and their exact solutions
W.I. FUSHCHYCH
In this paper we describe wide classes of nonlinear Schr?dinger-type PDEs which are
o
invariant under the Galilei group and its generalizations. We construct sets of ansatzes
for Galilei invariant equations, and exact classes solutions are found for some nonlinear
Schr?dinger equations.
o


1. Introduction
Let us consider the following nonlinear equations

L1 (?, ? ? ) ? S? ? F1 (x, ?, ? ? ),
(1)
p2 ? ?
S = p0 ? a , p0 = i , pa = ?i , a = 1, 2, 3,
2m ?x0 ?xa

?2??
+ F2 (x0 , x, ?, ? ? , ? , ? ? ) = 0,
L2 ? p0 ? + gab (x0 , x, ?, ? )
?xa ?xb 11
(2)
?? ?
?? ?
? ? ?(x0 , x1 , x2 , x3 ), x0 ? t, ? (x) = ,?= ,
?xa ?xa
1 1

where F1 , F2 , gab are some smooth functions,
L3 ? ? S? ? F3 (?, ? ? , ? , ? ? , ? , ? ? ),
1 1 2 2
(1 )
? 2 ??
2
?? ?
? (x) ? ? (x) ?
, .
?xa ?xb ?xa ?xb
2 2

In the present paper we consider the following problems.
Problem 1. Describe all nonlinear equations (1), (2) which are invariant with respect
to the Galilei group and its various generalizations.
Problem 2. Study the conditional symmetry of equation (l).
Problem 3. Construct classes of exact solutions for Galilei invariant equations.
The results of this talk have been obtained in collaboration with R. Cherniha,
V. Chopyk and M. Serov.

2. Galilei invariant quazilinear equations
Theorem 1 [1]. There are only three types of equations of the form (1)

(3)
S? = ?F (|?|)?,
In Proceedings of the International Symposium on Mathematical Physics “Nonlinear, Deformed
and Irreversible Quantum Systems” (15–19 August, 1994, Clausthal, Germany), Editors H.-D. Doebner,
V.K. Dobrev and P. Nattermann, Singapore – New Jersey – London – Hong Kong, World Scientific, 1995,
P. 214–222.
358 W.I. Fushchych

k ? R,
S? = ?|?|k ?, (4)
S? = ?|?|4/n ?, (5)
n = 1, 2, 3 . . . ,
which are invariant, correspondingly, with respect to the following algebras:
AG(1, n) = P0 , Pa , Jab , Ga , Q , a = 1, 2, . . . , n,
? ?
? p0 , Pa = ?i ? pa ,
P0 = i
(6)
?x0 ?xa
? ?
? ?? ?
Jab = xa pb ? xb pa , Ga = x0 pa ? mxa Q, Q=i ? ;
?? ??
AG(1, n) = AG(1, n), D ,
(7)
? ?
+ ?? ? ;
D = 2x0 p0 ? xa pa ? kI, k ? R, I=?
?? ??
AG(1, n) = AG1 (1, n), ? ,
(8)
m n
? = x2 p0 + x0 xa pa + x2 Q + x0 I,
0
2 2
? is arbitrary parameter, n is the number of space variables.
Note 1. If we put F = 0 in (1) we obtain the standard linear Schr?dinger equation
o
and its maximal invariance algebra is AG2 (1, n).
Corollary 1. There is only one nonlinear equation in the class of Schr?dinger equa-
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tions (1)
p2
p0 ? a ? = ?|?|4/n ? (9)
2m
which has the same symmetry as the linear Schr?dinger equation.
o
Let us answer the following question: whether there exist other equations in
the class (1) invariant under the Galilei algebra AG(1, n) but not invariant under
operators D and ? (7), (8).
The following theorem answers this question.
Theorem 2 [2]. There is only one equation of the form (1)
p2
? = ? ln(?? ? )?,
p0 ? a
(10)
? = ?1 + i?2
2m
which is invariant with respect to the following algebras:
AG3 (1, n) ? AG(1, n), B1 , B1 = I ? 2?1 x0 Q;
?1 = 0, ?2 = 0,
B2 = exp(2?2 x0 )(I + i?1 ??1 Q).
AG4 (2, n) = AG(1, n), B2 , ?3 = 0, 2

Note 2. The maximal invariance algebra of equation (10) with logarithmic nonlinearity
contains operators not admitted by the linear equation (1).
Corollary 2. Operators B1 , B2 generate the following transformations for ?:
? > ? = exp{(1 ? 2i?1 x0 )?1 }? for ?2 = 0,
? > ? = exp{?2 [2x0 ?2 (1 ? i?1 ??1 )}?, ?2 = 0,
2
? > ? = exp{?3 exp(2?1 x0 )}?, ?1 = 0, ?2 = 0,
where ?1 , ?2 , ?3 are group parameters.
Galilei invariant nonlinear Schr?dinger type equations
o 359

The following equation is widely used for description of dissipative systems
?? 1
?? ? i? ln(?(? ? )?1 )? + F2 (?? ? )?. (11)
i =
?t 2m
Equation (11) is usually called the phase Schr?dinger equation or the Schr?dinger–
o o
Langevin equation [4].
The main difference of equation (11) from equations (3), (4), (5), (10), (11) is that it
is not invariant with respect to the Galilei transformations. This equation does not the
standard Galilei relativity principle. However equation (11) has interesting symmetry
properties.
Theorem 3 [2]. The maximal invariance algebra of equation (11) is a 11-dimensional
Lie algebra

A = P0 , Pa , Jab , G(1) , Q ,
a
?m
G(1) = exp{2?x0 } pa + Q1 = exp{2?x0 }Q.
xa Q,
a
2
(1)
Corollary 3. Operators Ga generate the transformations

x0 > x0 = x0 , xa > xa = xa + exp{2?x0 }?a , (12)

? > ? = ? exp{i[?m exp(4?x0 )?2 + exp(2?x0 )xa ?a ]}, (13)

where ?2 = ?a ?a , ?a are group parameters.
(1)
So operators Ga as distinguished from the Galilei operators, generate nonlinear
transformations (12). In the first approximation by ? (12) coincides with the Galilei
transformations. It is known that the Galilei transformations are of the form
x0 > x0 = x0 ,
xa > xa = xa + x0 ?a ,
(14)
1
? > ? = exp im ?x + (?)2 x0 ?(x ).
2

3. Galilei invariant nonlinear equations
with first order derivatives
Let us consider equations

S? = F (x, ?, ? ? , ? , ? ? ). (15)
1 1

Theorem 4 [5]. There exist four classes of equations of the form (15) which are
invariant with respect to Galilei algebras:

S? = F1 (|?|, (?|?|)2 )?; (16)
AG1 (1, n) :

S? = |?|?2/k F2 (|?|?2+2/k (?|?|)2 )?, (17)
AG1 (1, n) :

S? = (?|?|)2 F3 (|?|)?; (18)

S? = |?|4/n F4 (|?|?2?4/n (?|?|)2 ). (19)
AG2 (1, n) :
360 W.I. Fushchych

Let us adduce some simplest G2 (1, 3) invariant equations:
S? = ?|?|4/3 ?, (20)

?|?|?|?|
S? = ?|?|2 (21)
?.
?xa ?xa

4. Conditional symmetry of the nonlinear Schr?dinger equation
o
Let us consider some nonlinear differential equation of s-th order:
(22)
L(x, ?, ? , ? , . . . , ? ) = 0,
s
1 2

? designates the set of all s-th order derivatives.
s
Let us assume that equation (22) is invariant with respect to a certain Lie algebra
A = X1 , X2 , . . . , Xn , where Xk are basis vectors of the algebra A.
This means that the following conditions must be satisfied:
(23)
Xk L = RL,
s

where Xk is the s-th prolongation of the operator Xk ? A, R = R(x, ?, ? , . . .) is some
s 1
differential expression.
Let us consider a set of operators which do not belong to the invariance algebra
of equation (22)
Yk ? A.
Y = Y1 , Y2 , . . . , Yr ,
Definition 1 [6, 7]. We shall say that equation (22) is conditionally invariant with
respect to the operators Y if there exists an additional condition

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