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algebra contains the Poincar? algebras AP (1, n + 1), AP (1, n) and so on, equation (2)

e

obeys the principle of Lorentz–Poincar?–Einstein relativity as well as Galilei relativity

e

(see [8] for more details on this e?ect).

There are other reasons for considering equation (2). First, (2) can be obtained as

a reduction of the hyperbolic equation

|?|2? ? ?2|?| = ???. (3)

Equation (3), with ? = m2 c2 / 2 was proposed by Vigier and Gueret [3] and by Guerra

and Pusterla [2] as an equation for de Broglie’s double solution [1]. Using the following

ansatz in (3)

? = ei(?? ?( x)/2)

u(?, ?x, ?x),

Preprint LiTH-MAT-R-97-11, Department of Mathematics, Link?ping University, Sweden, 1997,

o

12 p.

Some exact solutions of a conformally invariant nonlinear Schr?dinger equation

o 57

where ? = ?x = ?µ xµ and , ?, ?, ? are constant 4-vectors with ?2 = 2

= 0,

? 2 = ? 2 = ?1, ?? = ?? = ? = ? = 0, ? = 1, we obtain the equation

2 |u|

iu? + = u

2

|u|

2 2

? ?

with 2 = ?y2 + ?y2 , y1 = ?x, y2 = ?x. This is just equation (2). The ansatz described

1 2

above is used in reducing nonlinear complex wave equations to nonlinear Schr?dinger

o

equations (see [9] for more details).

A second reason for considering (2) is that it arises in connection with the so-called

classical limit of quantum mechanics ( > 0). Indeed, writing

? = A(t, x)ei?(t,x)/

in the free Schr?dinger equation

o

i?t = ? ?

2m

we obtain the system

??

2

1 A

?t (A2 ) + ? A2

(??)2 =

?t + , =0

2m 2m A m

> 0 gives

which, on taking the limit

??

1

?t (A2 ) + ? A2

(??)2 = 0,

?t + =0

2m m

which is the same system we obtain when we put u = Aei? into (2) (when m = 1/2).

It is thus possible to think of a classical particle having a wave-function u satisfying

(2), but we shall not pursue this interesting question here.

The main aim of our paper is to exploit the symmetry algebra AC(1, 4) to construct

exact solutions of equation (2) for n = 3. It is not yet possible to give a physical

interpretation of the solutions we obtain, but we believe that nontrivial solutions

of nonlinear equations are always of interest and give useful information about the

possible ?ows (trajectories, evolutions, bifurcations, asymptotics) of the dynamical

system described by (2). Of course, initial and boundary conditions will pick out

some special solutions of the equation which can be given a physical interpretation.

In order to construct solutions of (2) in explicit form, it is necessary to know all

inequivalent subalgebras of the algebra AC(1, 4), and then to construct corresponding

ansatzes which reduce (2) to equations in fewer independent variables, even ordinary

di?erential equations. It is not possible to realise this scheme (see [10] for details) in

full in this paper: we merely list those subalgebras of the extended Poincar? algebra

e

? (1, 4) = AP (1, 4), D which reduce (2) to ordinary di?erential equations which

AP

we are able to solve in general or ?nd particular solutions for. The solutions of these

ordinary di?erential equations give us exact solutions of (2).

58 P. Basarab-Horwath, L.L. Barannyk, W.I. Fushchych

2 Symmetry of (2) in terms of amplitude and phase

To simplify our work, it is convenient to go over to the amplitude-phase representation

of the function u:

u(t, x) = A(t, x)ei?(t,x) = eR(t,x)+i?(t,x)

in terms of which equation (2) becomes:

?t + ?? · ?? = 0, (4)

? + 2?? · ?R = 0. (5)

Rt +

Using the standard algorithm for calculating Lie point symmetries (see, for examp-

le, [11, 13, 12, 8]) we ?nd the following result:

Theorem 1. The maximal point-symmetry algebra of the system of equations (4), (5)

is algebra with basis vector ?elds

1

Pn+1 = v (2?t ? ?? ), N = ?R , Jab = xa ?b ? xb ?a ,

P t = ?t , Pa = ?a ,

22

1 1

J0,n+1 = t?t ? ??? , J0a = v xa ?t + (t + 2?)?xa + xa ?? ,

2

2

1 1

Ja,n+1 = v ?xa ?t + (t ? 2?)?xa + xa ?x0 ,

2

2

n

D = ? t?t + xa ?a + ??? ? ?R ,

2

v 2

x2

x n

+ 2?2 ?? ? (t + 2?)?R ,

K0 = 2 t + ?t + (t + 2?)xa ?xa +

2 4 2

v 2 2

x x n

Kn+1 = ? 2 t ? ?t +(t ? 2?)xa ?xa + ? 2?2 ?? ? (t ? 2?)?R ,

2 4 2

Ka = 2xa D ? (4t? ? x2 )?xa .

The above algebra is equivalent to the extended conformal algebra AC(1, n + 1) ?

N . In fact, with new variables

1 1

x0 = v (t + 2?), xn+1 = v (t ? 2?) (6)

2 2

the operators in Theorem 1 can be written as

J?? = x? ?? ? x? ?? , N = ?R ,

P ? = ?? ,

(7)

n

D = ?x? ?? + N, K? = ?2x? D ? (xµ xµ )?? .

2

Remark 1. It follows from Theorem 1 that the nonlinear Schr?dinger equation (2) is,

o

in 1 + 3 time-space, invariant with respect to the Poincar? group P (1, 4) of 1 + 4 time-

e

space. The basis elements of the algebra AP (1, 4) are P0 , P1 , P2 , P3 , P4 , J?? , J04 , J4a .

We also have that the new “time” x0 and the new coordinate x4 in (6) depend linearly

on the phase function ? and on t, the time.

Some exact solutions of a conformally invariant nonlinear Schr?dinger equation

o 59

Subalgebras of AP (1, 4) ? N :

3

ansatzes and solutions

In this section we exploit those subalgebras of the algebra AP (1, 4)? N which reduce

the equation (2) in 1+3 time-space dimensions to ordinary di?erential equations which

we are able to solve. In fact, we use the system (4), (5), since we construct ansatzes for

the functions ? and R which, when substituted into (4) and (5), yield exact solutions

of (2).

Using the methods exposed in [15, 10], we have made a detailed subalgebra analysis

of AP (1, 4) ? N , and we have described all inequivalent subalgebras of rank 3. Here,

we give a list of these algebras, the ansatzes and the exact solutions obtained.

A1 = J12 + dN, P3 + N, P4 (d ? 0)

Ansatz:

1 x2

? = ? t + f (?), R = x3 ? d arctan ? = x2 + x2 .

+ g(?), 1 2

2 x1

Solution:

x2 + x2

1

? =? t+? + C1 , ? = ±1,

1 2

2 2

x2 1

R = x3 ? d arctan ? ln(x2 + x2 ) + C2 ,

1 2

x1 4

where C1 , C2 are constants. These solutions describe processes which have phase

linear in time and amplitude constant in time, linear in x3 .

A2 = J04 + dN, P1 + N, P2

Ansatz:

1

?= f (?), R = ln t + x1 + g(?), ? = x3 .

t

Solution:

1 2d + 1

R = d ln t + x1 ? ? ln |x3 + C1 | + C2 , ? = ±1.

(x3 + C1 )2 ,

?=

4t 2

A3 = J04 + d1 N, J12 + d2 N, P3 + d3 N

Ansatz:

1 x2

R = d1 ln t ? d2 arctan

? = x2 + x2 ,

?= f (?), + d3 x3 + g(?),

1 2

t x1

where d1 , d2 , d3 are constants.

Solution:

( x2 + x2 + C1 )2

1 2

?= ,

4t

x2 1

R = d1 ln t ? d2 arctan + d3 x3 ? ln x2 + x2 ?

1 2

x1 4

1

? d1 + x2 + x2 + C1 + C2 .

ln 1 2

2

60 P. Basarab-Horwath, L.L. Barannyk, W.I. Fushchych

A4 = J04 + dN, J23 , P2 , P3

Ansatz:

1

? = f (?), R = d ln |t| + g(?), ? = x1 .

t

Solution:

(x1 + C1 )2 1

R = d ln |t| ? d + ln |x1 + C1 | + C2 .

?= ,

4t 2

A5 = G1 , J04 + d1 N, P3 + d2 N with d1 arbitrary and d2 = 0, 1

Ansatz:

x2 1

R = d1 ln |t| + d2 x3 + g(?),

1

?= + f (?), ? = x2 .

4t t

Solution:

x2 + (x2 + C1 )2

R = d1 ln |t| + d2 x3 ? (d1 + 1) ln |x2 + C1 | + C2 .

?= 1 ,

4t

A6 = J12 , J13 , J23 , P4 + dN (d = 0, 1)

Ansatz:

v

1

? = ? t + f (?), R = ?d 2t + g(?), ? = x2 + x2 + x2 .

1 2 3

2

Solution:

x2 + x2 + x2

1

? =? t+? + C1 , ? = ±1,

1 2 3

2 2

v 1

R = ?d 2t + d? x2 + x2 + x2 ? ln(x2 + x2 + x2 ) + C2 .

1 2 3

1 2 3

2

A7 = G1 , G2 , J04 + d1 N, J12 + d2 N

Ansatz:

x2 + x2 1 x2

R = d ln |t| ? d2 arctan

1 2

?= + f (?), + g(?), ? = x3 .

4t t x1

Solution:

x2 + x2 + (x3 + C1 )2

1 2

?= ,

4t

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