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Ga = x0 ?a + ??j a , J0a = xa ?0 + x0 ?a + j a ?? + ??j a ,

D(1) = 2x0 ?0 + xa ?a ? n??? ? (n + 1)j a ?j a , D(2) = xµ ?µ ? n??? ? nj a ?j a ,

A = x2 ?0 + x0 xa ?a ? nx0 ??? + (xa ? ? (n + 1)x0 j a )?j a ,

0

Kµ = 2xµ D(2) ? x? x? gµi ?i ? 2x? Sµ? , Sµ? = gµi j ? ?j i ? g?i j µ ?j i ,

?

? 1, µ = ? = 0,

gµ? = ?1, µ = ? = 0, µ, ?, i = 0, 1 . . . , n; a, b = 1, 2, . . . , n.

?

0, µ = ?,

Corollary 3. The continuity equation satis?es the Galilei relativity principle as well

as the Lorentz–Poincar?–Einstein relativity principle.

e

Thus, depending on the de?nition of ? and j, we come to di?erent quantum

mechanics.

2. Let us consider the scalar complex–valued wave functions and de?ne ? and j in

the following way

? = f (uu? ),

?u? ??(uu? ) (5)

1 ?u ?

j k = ? ig(uu? ) u ?u + , k = 1, 2, . . . , n.

2 ?xk ?xk ?xk

where f , g, ? are arbitrary smooth functions, f = const, g = 0. Without loss of

generality, we assume that f ? uu? .

Let us describe all functions g(uu? ), ?(uu? ) for continuity equation (1), (5) to be

compatible with the Galilei relativity principle, de?ned by the following transforma-

tions:

t > t = t, xa > xa = xa + va t.

Here we do not ?x transformation rules for the wave function u.

Theorem 2. If ? and j are de?ned according to formula (5), then the continuity

equation (1) is Galilei-invariant i?

?u? ??(uu? )

1 ?u ?

?

j =? i u ?u

k

(6)

? = uu , + , k = 1, 2, . . . , n.

2 ?xk ?xk ?xk

The corresponding generators of Galilei transformations have the form

Ga = x0 ?a + ixa (u?u ? u? ?u? ) , a = 1, 2, . . . , n.

If in (6)

? = ?uu? , (7)

? = const,

then the continuity equation (1), (6), (7) coincides with the Fokker–Planck equation

??

+ ? · j + ??? = 0, (8)

?t

Continuity equation in nonlinear quantum mechanics 139

where

?u?

1 ?u ?

?

j =? i u ?u

k

(9)

? = uu , , k = 1, 2, . . . , n.

2 ?xk ?xk

The continuity equation (1), (6), (7) was considered in [2, 6].

Let us investigate the symmetry of the nonlinear Schr?dinger equation

o

??(uu? )

1

u = F uu? , (?(uu? ))2 , ?(uu? ) u, (10)

iu0 + ?u + i ?

2 2uu

where F is an arbitrary real smooth function.

For the solutions of equation (10), equation (1), (6) is satis?ed and is compatible

with the Galilei relativity principle. Schr?dinger equations in the form of (10), when

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? ?

?(uu ) = ?uu for ?xed function F , were considered in [1–8].

In terms of the phase and amplitude u = R exp(i?) , equation (10) has the form

1 1

R0 + Rk ?k + R?? + ?? = 0,

2 2R (11)

1 1 2

?0 + ?2 ? ?R + F R2 , ? R2 2

, ?R = 0.

k

2 2R

Theorem 3. The maximal invariance algebras for system (11), if F = 0, are the

following:

(12)

1. Pµ , Jab , Q, Ga , D

when ? is an arbitrary function;

(13)

2. Pµ , Jab , Q, Ga , D, I, A

when ? = ?R2 , ? = const.

In (12) and (13) we use the following designations:

Pµ = ?µ , Jab = xa ?xb ? xb ?xa , a < b,

Ga = x0 ?xa + ixa ?? , Q = ?? , D = 2x0 ?x0 + xa ?xa , I = R?R ,

(14)

n 1

A = x2 ?x0 + x0 xa ?xa ? x0 R?R + x2 ?? ,

0

2a

2

µ = 0, 1, . . . , n; a, b = 1, 2, . . . , n.

Algebra (13) coincides with the invariance algebra of the linear Schr?dinger equation.

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Corollary 4. System (11), (7) is invariant with respect to algebra (13) if

R?R

F = R?1 ?R N ,

(?R)2

where N is an arbitrary real smooth function.

3. Let us consider a more general system than (10)

1

(15)

iu0 + ?u = (F1 + iF2 )u,

2

140 W.I. Fushchych, V.M. Boyko

where F1 , F2 are arbitrary real smooth functions,

Fm = Fm uu? , (?(uu? ))2 , ?(uu? ) u, (16)

m = 1, 2.

The structure of functions F1 , F2 may be described in form (16) by virtue of

conditions for system (15) to be Galilei-invariant.

In terms of the phase and amplitude, equation (15) has the form

1 1 1

R0 + Rk ?k + R?? ? RF2 = 0, ?0 + ?2 ? (17)

?R + F1 = 0,

k

2 2 2R

2

where Fm = Fm R2 , ? R2 , ?R2 , m = 1, 2.

Theorem 4. System (17) is invariant with respect to the generalized Galilei algebra

AG2 (1, n) = Pµ , Jab , Ga , Q, D, A if it has the form

(?R)2

1 ?R

R0 + Rk ?k + R?? ? R1+4/n M ; 1+4/n = 0,

R2+4/n R

2

(?R)2

1 1 ?R

?0 + ?2 ? ?R + R4/n N ; 1+4/n = 0,

k

R2+4/n R

2 2R

where N , M are arbitrary real smooth functions. The basis operators of the algebra

AG2 (1, n) are de?ned by (14) and D = D ? n I.

2

Theorem 5. System (17) is invariant with respect to algebra (13) if it has the form

1 R?R

R0 + Rk ?k + R?? ? ?R M = 0,

(?R)2

2

(18)

1 1 ?R R?R

?0 + ?2 ? ?R + N = 0,

k

(?R)2

2 2R R

where N , M are arbitrary real smooth functions.

System (18) written in terms of the wave function has the form

|u|?|u| |u|?|u|

1 ?|u|

(19)

iu0 + ?u = N + iM u.

|u| (?|u|)2 (?|u|)2

2

Equation (19) is equivalent to the following equation

?(uu? ) (uu? )?(uu? ) (uu? )?(uu? )

1 ? ?

iu0 + ?u = N + iM u.

(uu? ) (?(uu? ))2 (?(uu? ))2

2

Thus, equation (18) admits an invariance algebra which coincides with the inva-

riance algebra of the linear Schr?dinger equation with the arbitrary functions M , N .

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Remark 1. With certain particular M and N the symmetry of system (18) can be

essentially extended. E.g., if in (18) N = 1 , then the second equation of the system

2

(equation for the phase) will be the Hamilton–Jacobi equation [5].

Let us consider some forms of the continuity equation (1) for equation (18).

Continuity equation in nonlinear quantum mechanics 141

Case 1. If M = 0, then for solutions of equation (18) equation (1) holds true,

where the density and current can be de?ned in the classical way (9).

(?R)2

Case 2. If ?R M = ?? ?R + , then for solutions of equation (18), the

R

continuity equation (1), (6), (7) (or the Fokker–Planck equation (8), (9)) is valid.

Case 3. If M is arbitrary then for solutions of equation (18), the continuity equation

is valid, where the density and current can be de?ned by the conditions

?u? |u|?|u|

? 1 ?u ?

? = uu? , ?·j = ?i u ?u ? 2|u|?|u| M .

(?|u|)2

?xk 2 ?xk ?xk

Thus, we constructed wide classes of the nonlinear Schr?dinger-type equations

o

which is invariant with respect to algebra (13) (maximal invariance algebra of the

linear Schr?dinger equation) and for whose solutions the continuity equation (1) is

o

valid.

Acknowledgements. The authors would like to thank the INTAS, SOROS,

DKNT of Ukraina foundations for ?nancial support. V. Boyko also would like to

thank the AMS for ?nancial support.

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nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993, 436 p.

2. Doebner H.-D., Goldin G.A., Properties of nonlinear Schr?dinger equations associated with

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di?eomorphism group representations, J. Phys. A: Math. Gen., 1994, 27, 1771–1780.

3. Fushchych W.I., Cherniha R.M., Galilei-invariant nonlinear equations of the Schr?dinger-type

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and their exact solutions I, II, Ukr. Math. J., 1989, 41, 1349–1357, 1687–1694.

4. Fushchych W.I., Cherniha R.M., Galilei-invariant systems of nonlinear systems of evolution

equations, J. Phys. A: Math. Gen., 1995, 28, 5569–5579.

5. Fushchych W., Cherniha R., Chopyk V., On unique symmetry of two nonlinear generalizations

of the Schr?dinger equation, J. Nonlinear Math. Phys., 1996, 3, ¹ 3–4, 296–301.

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6. Fushchych W.I., Chopyk V., Nattermann P., Scherer W., Symmetries and reductions of nonlinear

Schr?dinger equations of Doebner–Goldin type, Reports on Math. Phys., 1995, 35, ¹ 1, 129–138.

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7. Vigier J.-P., Particular solutions of a nonlinear Schr?dinger equation carrying particle-like sin-

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gularities represent possible models of de Broglie’s double theory, Phys. Lett. A, 1989, 135,

99–105.

8. Schuh D., Chung K.-M., Hartman H.N., Nonlinear Schr?dinger-type ?eld equation for the

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description of dissipative systems, J. Math. Phys., 1984, 25, ¹ 4, 786.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 142–158.

Higher symmetries and exact solutions

of linear and nonlinear Schr?dinger equation

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W.I. FUSHCHYCH, A.G. NIKITIN

A new approach for the analysis of partial di?erential equations is developed which

is characterized by a simultaneous use of higher and conditional symmetries. Higher

symmetries of the Schr?dinger equation with an arbitrary potential are investigated.

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Nonlinear determining equations for potentials are solved using reductions to Wei-

erstrass, Painlev?, and Riccati forms. Algebraic properties of higher order symmetry

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