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356 W.I. Fushchych

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mathematical physics, Kluwer Academic Publishers, 1993, 430 p. (Russian version, 1989, Naukova

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the Schr?dinger group, Theor. and Math. Fisika, 1983, 56, ¹ 3, 387–394.

o

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Ukrainian Acad. Sci., 1990, ¹ 7, 24–28.

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tion, Dopovidi of the Ukrainian Acad. Sciences, 1991, ¹ 8, 23–26.

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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

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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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