ñòð. 94 |

(37a) is the heat equation with spatial source of energy absorption (extraction) q =

U f (V ), proportional to the temperature U, with an additional condition of elliptical

equation (37b) being imposed on proportionality coeficient f (V ) (in particular we can

consider f (V ) = V ). Thus we have obtained a class of nonlinear heat equations with

an additional condition for the source that preserve Galilean symmetry of the linear

heat equation. This result is quite non-trivial, since it is well-known fact that among

nonlinear heat equations with a source

Ut = ?U + q(U )

not a single one is invariant with respect to Galilei algebra AG(1, n) [3]. As it is seen,

this “symmetry contradiction” between the linear and nonlinear heat equations can be

solved in two ways: either the source is supposed to depend explicitly on temperature

and independent variables t, x1 , . . . , xn [3], or an additional condition equation (37b)

upon the source is imposed as above.

It should be noted that in case f = ?1 V 2/?2 , g = ?2 V 1+2/?2 , 0 = ?2 , ?k ? R

system (37) is invariant under AG1 (1, n) algebra (2a)–(2c). If the system (37) has

the form

Ut = ?U + ?1 U V 4/n , (38a)

0 = ?V + ?2 V 1+4/n , (38b)

Galilei-invariant nonlinear systems of evolution equations 391

it is invariant under AG2 (1, n) algebra with basic operators (2) for ?2 = 0, ?1 = 1,

i.e. heat equation (38a) with nonlinear condition (38b) for the source conserves all

the non-trivial Lie symmetry of the linear heat equation

Ut = ?U.

Note 3. If V is a fixed given function on independent variables t, x1 , . . . , xn , equation

(38a) can lose any symmetry.

In conclusion, the interesting system of the form (33) should be considered,

namely

?Ut = ?U + ?1 U 2 V ?1 ,

(39)

?Vt = ?V + ?2 U, ?1 = ?2 .

Theorem 5. The maximal algebra of invariance for the system (39) is the generali-

zed Galilei algebra with the basic operators (2a), (2b) and

n ?2

D = 2tPt + xa Pa ? 2U ?U ? + Q? ,

?1 ? ?2

2

1 ?

? = ?t2 Pt + tD ? |x|2 Q? ? V ?U .

?1 ? ?2

4

By the way, among the systems of the form (33) in case where ?2 = ?1 = ? there

is not an AG2 (1, n)-invariant system in the standard representation (2). Note that the

system (39) can be considered as a particular case of the conservation equations for

normal and mutant cells [7, 24].

Some classes of exact solutions for the system (39) are obtained in [25].

Acknowledgements

The authors acknowledge financial support by DKNT of Ukraine (project No

11.3/42) and WIF by a Soros Grant.

1. Fushchych W.I., Cherniha R.M., Ukr. Math. J., 1989, 41, ¹ 10, 1161–1167; ¹ 12, 1456–1463.

2. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of nonli-

near mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.

3. Fushchych W.I., Cherniha R.M., J. Phys. A, 1985, 18, 3491–3503.

4. Aris R., The theory of reaction and diffusion in permeable catalyst, Oxford, 1975.

5. Wilhelmsson H., Ukr. Phys. J, 1993, 38, ¹ 1, 44–53; Physica Scripta, 1992, 46, 177–181.

6. Fife P.C., Mathematical aspects of reacting and diffusing systems, Berlin, Springer, 1979.

7. Murray J.D., Mathematical biology, Berlin, Springer, 1989.

8. Doebner H.-D., Goldin G.A., Phys. Lett. A, 1992, 162, 397–407.

9. Goldin G.A., Svetlichny G., J. Math. Phys., 1994, 35, 3322–3332.

10. Malomed B.A., Steflo L., J. Phys. A, 1991, 24, L1149–L1153.

11. Niederer U., Helv. Phys. Acta, 1972, 45, 808–816.

12. Fushchych W.I., Cherniha R.M., On exact solutions of two multidimensional nonlinear Schr?dinger-

o

type equations, Preprint ¹ 86.85, Kiev, Institute of Mathematics of Ukrainian Acad. Sci., 1986.

13. Fushchych W.I, Cherniha R.M., Proc. Ukr. Acad. Sci., 1993, ¹ 8, 44–51.

14. Zakharov V., Shabat B., Zhurn. Eksper. Teor. Fiz. (JETP), 1971, 61, ¹ 1, 118–134.

15. Fushchych W.I., Moskaliuk S.S., Lett. Nuovo Cimento, 1981, 31, ¹ 16, 571–576.

392 W.I. Fushchych, R.M. Cherniha

16. Clarkson P.A., Cosgrove C.M., J. Phys. A, 1987, 20, 2003–2024.

17. Gagnon L., Winternitz P., J. Phys. A, 1988, 21, 1493–1511.

18. Gagnon L., J. Phys. A, 1992, 25, 2649–2667.

19. Fushchych W.I., Serov N.I., J. Phys. A, 1987, 20, L929–L933.

20. Kundu A., J. Math. Phys., 1984, 25, 3433–3438.

21. Calogero F., De Lillo S., Inverse Problems, 1988, 4, L33–L37.

22. Cherniha R.M., Ukr. Phys. J., 1995, 40, ¹ 4, 376–384.

23. Boyer C.P., Penafiel M.N., Nuovo Cimento B, 1976, 31, ¹ 2, 195–210.

24. Sherratt J.A., Physica D, 1994, 70, 370–382.

25. Cherniha R.M., On exact solutions of a nonlinear diffusion-type system, in Symmetry Analysis and

Exact Solutions of Equations of Mathematical Physics, Kyiv, Institute of Mathematics of Ukrainian

Acad. Sci., 1988, 49–53.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 393–401.

Symmetries and reductions

?

of nonlinear Schrodinger equations

of Doebner–Goldin type

W.I. FUSHCHYCH, V. CHOPYK, P. NATTERMANN, W. SCHERER

We compute symmetry algebras for nonlinear Schr?dinger equations which contain an

o

imaginary nonlinearity as derived by Doebner and Goldin and certain real nonlinearities

not depending on the derivatives. In the three-dimensional case we find the maximal

symmetry algebras for equations of this type. Admitting other imaginary nonlinearities

does lead to similar symmetry algebras. These symmetries are used to obtain explicit

solutions of these equations by means of reduction.

1. Introduction

Recently, a new nonlinear Schr?dinger equation as the evolution equation of a

o

quantum mechanical system on R has been derived from general principles by

n

Doebner and Goldin [1–4]. Their derivation is based on the representation theory

of the semidirect product of the group of diffeomorphisms with the smooth functi-

ons on Rn and results in the replacement of the usual continuity equation ? = ??j

?

? and j = ??? ? ???)) associated with the linear Schr?dinger

?

(where ? = ?? o

2mi (?

equation by the Fokker–Planck equation ? = ??j + d?? describing diffusion of the

?

probability density ?. This Fokker–Planck equation for the probability density can be

derived from a nonlinear Schr?dinger equation which has to be of the form

o

d ??

? ?

? (1)

i ?= ?+V +i ? + F [?, ?] ?,

2m 2?

where F is assumed to be an arbitrary real functional. Doebner and Goldin proceeded

?

with the requirement that F [?, ?] should have similar properties as the imaginary

nonlinear functional, and were thus led to a five parameter functional including deri-

vative terms [4]. Galilei-invariant nonlinear Schr?dinger equations of type (1), where

o

d = 0 and F depends on the wave function and its first order derivatives, were

described by Fushchych and Cherniha [5].

On the other hand, equations similar to (1) have been considered in plasma phy-

?

sics [6] and for d = 0 and F [?, ?] = a? it reduces to the usual nonlinear Schr?dinger

o

equation which appears in many subfields of physics. It seems therefore worthwhile to

investigate the Lie symmetries for equations of this type and to use them to construct

solutions. This is what we shall do in this paper.

Obviously, we shall have to restrict the functional F suitably since otherwise it

would be impossible to say anything at all about the symmetries of this equation.

Whereas the maximal Lie symmetry of the Doebner–Goldin equation has already

been calculated [7], we shall restrict our considerations in this paper to another class

of functional F given by (sufficiently smooth) functions f of a single real variable:

? (2)

F [?, ?] := f (?),

Rep. on Math. Phys., 1995, 35, P. 129–138.

394 W.I. Fushchych, V. Chopyk, P. Nattermann, W. Scherer

which includes many physically interesting models [8, 9]. Although we leave the

framework set by Doebner and Goldin if f is not real, we will consider a slightly

morel general case of complex valued functions f since calculations are similar. For

d = 0 the Lie symmetry of this nonlinear Schr?dinger equation has been discussed in

o

[10, 11, 12].

In Section 2 we will determine the maximal Lie symmetries of the nonlinear

Schr?dinger equations (1) with functional of type (2). It turns out that the most

o

prominent cases, i.e. f (p) ? ?k and f (?) ? ln ?, admit the largest symmetry algebras.)

Subalgebras of the maximal symmetry algebras will be used in Section 3 to reduce

equation (1) and find exact solutions. We close this paper with some further remarks)

on the equations and the solutions obtained.

2. Lie symmetry algebra

2.1. n ? 3. First, we shall treat the physically most interesting case of three space

dimensions (n = 3) for which we will determine the maximal Lie symmetry algebra

of equation (1) with the complex valued functional (2). In order to do so, we write ?

in terms of an amplitude function R and a phase function S:

?(x, t) = R(x, t)eiS(x,t) .

With the decomposition of f into the real and imaginary parts, f = u+iv, equation (1)

is thus equivalent to two real evolution equations:

(?R)2

R?S + 2?R · ?S ? d ?R + ? Rv(R2 ) = 0, (3)

?t R +

2m R

?R

(?S)2 ? + u(R2 ) = 0. (4)

?t S +

2m R

Vector fields acting on the space of independent (x1 , x2 , x3 , t) and dependent (R, S)

variables

X = ?j ?xj + ? ?t + ??R + ??S ,

are generators of a Lie symmetry of the equations (3) and (4), if the coefficients

?j , ? , ?, ? satisfy the so-called determining equations. A detailed description of

the theory can be found in the monographs [10, 13, 14]. Since the procedure is

purely algorithmic, we use a Mathematica program [15] to obtain these equations.

This leads to 62 determining equations among which only two contain the real and

imaginary part of f . These two equations determine the functional F of equation (1).

The integration of the 60 remaining equations yields the following coefficients of the

vector field X:

?j = (2c1 t + c2 )xj + wjl xl + vj t + aj ,

? = 2c1 t2 + 2c2 t + 2c3 ,

(5)

? = ?(t)R,

m

? = (c1 x2 + vk xk ) + ?(t),

Symmetries and reductions of nonlinear Schr?dinger equations

o 395

where ci , vj and aj are real constants, wjl is an antisymmetric matrix with real

constant coefficients, and ? and ? are real functions of time. The two remaining

determining equations which contain the functions u and v thus read

1

?(t)R2 u (R2 ) + (2c1 t + c2 )u(R2 ) + ? (t) = 0, (6)

2

1

?(t)R2 v (R2 ) + (2c1 t + c2 )v(R2 ) ? (? (t) + 2nc1 ) = 0. (7)

2

For the cases n = 1, 2 the resulting equations are exactly the same, with the

understanding that in equation (7) the dimension n has to be inserted. In order

to calculate the maximal symmetry, we solve the ordinary differential equation (7)

for ? and then (6) for ?, requiring that the resulting functions do not depend on R.

Neglecting the case of constant functions u = C — which can be transformed to zero

by the map ? ?> eiCt ? — this leads to the following six possible cases.

1. For arbitrary functions u and v one has to require that their coefficients and the

inhomogeneous terms in equations (6) and (7) vanish, which leaves only the centrally

extended Galilei algebra g(n = 3) = H, Pj , Jjk , Gj , Q with ten generators:

Jjk = xj ?xk ? xk ?xj ,

H = ?t , Pj = ?xj ,

(8)

m

G = t?xj + xj ?S , Q = ?S .

2. A larger algebra is obtained if u and v are of the from

u(R2 ) = ?1 R2k , v(R2 ) = ?2 R2k ,

in which case equations (6) and (7) reduce to linear inhomogeneous equations in u

and v, respectively. Requiring the coefficients and the inhomogeneous term to vanish

allows the maximal Lie symmetry to contain an additional generator

1

D = 2t?t + xk ?xk ? (9)

R?R ,

k

and this algebra H, Pj , Jjk , Gj , Q, D has been named the Galilei similitude algeb-

ra [16]. D generates the dilations.

3. Calculations of the previous case show that the Lie symmetry has an extra

generator if k = n = 1 :

1

3

m2

x ?S ? ntR?R ,

C = t2 ?t + txk ?xk + (10)

2

yielding the maximal Lie symmetry algebra of the free linear Schr?dinger equa-

o

tion [17] H, Pj , Jjk , Gj , Q, D, C (Schr?dinger algebra). The transformations genera-

o

ted by C are called projective or conformal transformations.

4. If u(R2 ) = ?1 ln(R2 ) and v = ?3 is a constant, we obtain the maximal Lie

symmetry algebra H, Pj , Jjk , Gj , Q, D, B , where

B = R?R ? 2?1 t?S . (11)

Note that for nonvanishing ?1 the constant ?3 can be transformed to zero by the map

2

? ?> e??3 t+i?1 ?3 t ?.

396 W.I. Fushchych, V. Chopyk, P. Nattermann, W. Scherer

5. If u(R2 ) = ?1 ln(R2 ) and v(R2 ) = ?2 ln(R2 ) + ?3 with ?2 = 0, equation (7)

ñòð. 94 |