ñòð. 21 |

exp[i(V · y/2 ? tV 2 /4 ? ta2 )]

v=A ,

cosh(a · y ? a · V t)

and

exp[i(V · y/2 ? tV 2 /4 ? ta2 )]

v=A ,

sinh(a · y ? a · V t)

as solutions of (22), with V = (V1 , V2 ) an arbitrary vector. From this we ?nd that

the hyperbolic equation

2|?|µ |?|µ

2? = ? ?

|?|2

admits the solutions

exp[i(V1 (?x)/2 + V2 (?x)/2 ? V 2 (?x)/4 ? ?x/2 ? a2 (?x))]

?=A

cosh(a1 (?x) + a2 (?x) ? (a · V )(?x))

92 P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

and

exp[i(V1 (?x)/2 + V2 (?x)/2 ? V 2 (?x)/4 ? ?x/2 ? a2 (?x))]

?=A .

sinh(a1 (?x) + a2 (?x) ? (a · V )(?x))

Note that we have only used two Galilean boosts to obtain these two-parameter

families of solutions. We can introduce more parameters by using the other symmetries

of the hyperbolic equation and the corresponding Schr?dinger equations.

o

A third example is the hyperbolic equation

?µ ?µ

2? = 2p|?|2 ? ? C (23)

?

with C = 1. Using the ansatz

? = e?i(?x)/2(1+C) v(?x, ?x, ?x)

is straightforward to show that (23) reduces to the equation

va v a

ivt + ?v + 2p|v|2 v = ?C (24)

.

v

In 1+1 space-time, equation (24) is the Malomed–Sten?o equation [32] in plasma

physics which admits solitons. Equation (24) admits the solution

v = A sech (n · y) exp(i(C + 1)n2 t)

(which in 1+1 dimensions is the Malomed–Sten?o soliton), where A2 = n2 (C + 2)/2p

and n = (n1 , n2 ) is an arbitrary vector. We can now act on this solution with the

Galilean boosts

iya

(v?v ? v ?v ),

Ga = t?a + ??

2(1 + C)

which are symmetries of (24), and we obtain

V ·y V 2t

v = A sech (n · y ? n · V t) exp i (C + 1)n t + ?

2

2(1 + C) 4(1 + C)

as a two-parameter family of solutions of (24). We are then able to construct the

following solution of (23):

? V (?x)

2

V1 (?x) V2 (?x)

exp i (1 + C)n2 (?x) ? ?x

+ +

2(1+C) 2(1+C) 2(1+C) 4(1+C)

?=A .

cosh(n1 (?x) + n2 (?x) ? (n · V )(?x))

5 Conclusions

These are just some examples of hyperbolic equations which reduce down to nonlinear

Schr?dinger equations. There are of course more. For instance, the hyperbolic equation

o

2|?|

2? = ? ? ??, (25)

|?|

Solutions of the relativistic nonlinear wave equation 93

which arises in the context of de Broglie’s double solution [33, 1], reduces, with our

ansatz, to

?|v|

i?t v = ??v + (26)

v + ?v;

|v|

an equation which was considered by Guerra and Pusterla [34] in the context of

a nonlinear Schr?dinger equation. The terms 2|?|/|?| and ?|v|/|v| are called the

o

quantum potentials [1]. Both equations (25) and (26) are conformally invariant, (25)

being invariant under the conformal algebra AC(1, n+2), and (26) under AC(1, n+1)

in n + 1 space-time dimensions (see [40]). These remarkable symmetry properties are

due to the quantum potential term. They share this symmetry with a wide class of

other equations [36, 37].

Despite this connection, we are as yet unable to give a clear physical meaning

to the reduction and the ansatz, other than the purely Lie-algebraic one. That we

should expect some sort of physical interpretation is suggested by the use of complex

hyperbolic equations by Grundland and Tuszynski in [10] in the context of super?ui-

dity and liquid crystal theory.

It is also natural to ask if it is possible to obtain a nonlinear complex hyperbolic

wave equation from a Schr?dinger equation. It is, of course, not possible from an

o

equation of the form

ivt + ?v = F (|v|)v.

However, if we consider

?2v ?2v

? 2 = F (|v|)v,

ivt +

?x2 ?y

and put

v = ei(x+y) w(x ? 2t, y + 2t),

then we ?nd that w satis?es the equation

?2w ?2w

? = F (|w|)w,

?? 2 ?? 2

with ? = x ? 2t, ? = y + 2t. It thus seems of interest to investigate equations of the

type

??

+ 2? = F (|?|)?.

i

?t

This type of equation is also of interest in quantum physics: the equation

?? 1

(2 ? m2 )?

i =

?t 2m

(with interaction terms involving the electromagnetic potential) was used by Fock as

an analogue of the Hamilton–Jacobi equation in quantum mechanics, where t was

interpreted as the proper time (see [38] for more details on parametrized relativistic

quantum theories). Feynman in [39] considered the equation

?? 1

= (?µ ? eAµ )(? µ ? eAµ )?.

i

?t 2

94

Table 1. Reduction to ordinary di?erential equations. These ansazes and reduced equations are for equations (9), (10) and (11).

Subalgebra Ansatz ? Reduced equation

?

v = exp(?i?t)?(?) ? + a? = ?F (|?|)?

y1

P2 , T + 2aM (a ? R)

a2

y1 2 2 ? ?

?(?)

v = exp i bt + a arctan ? = ?F (|?|)?

J12 + 2aM, T ? 2bM (a, b ? R) 4? ? + 4? ? b +

y1 + y2

?

y2

a2 t 3 ay1 t

?(?)

+ 4? +

? a? ? = ?F (|?|)?

T + aG1 , P2 (a > 0) v = exp i ? at2 ? 2y1

6 2 4

2

1

iy1 ?

?(?) t

v = exp 4? +

G1 , P2 ? = ??F (|?|)?

4t 2?

2

y2 1 1

y1 ?1

?(?) t

v = exp i +2 ?+ +

G1 + aP1 , G2 (a ? R) ? = ??F (|?|)?

?

4t

4(t ? a) 2 ??a

?

t

v = ?(?) i? = ?F (|?|)?

P1 , P 2

Table 2. Reduction by two-dimensional subalgebras of AG1 (1, 2). These ansazes and reduced equations are for equations (10) and (11).

Subalgebra Ansatz ? Reduced equation

2

i

y1 ? ?

v = t?(ia+1/k) ?(?) = ?|?|k ?

P2 , D + 4aM (a ? R) 4? ? + (2 ? i?)? + a ?

t k

2 2

i i

?(ia+1/k) iy2 y1 ? ?

?(?)

v=t exp = ?|?|k ?

G2 , D + 4aM (a ? R) 4? ? + (2 ? i?)? + a + ?

t k

4t 2

2

y1

?(ia+2/k) ? ?

?(?) (? 2 + 1)? + 2? ia + + 1 ? +

T, D + 2aM (a ? R) v = y1

k

y2

2 2

+ ia + ia + + 1 ? = ?|?|k ?

k k

2 2

y1 y1 + y2 ? ?

?(?)

v = t?(ib+1/k) exp ia arctan

J12 + 2aM, D + 4bM 4? ? + (4 ? i?)? ?

t

y2

a2 i

(a ? 0, b ? R) ? ?b+ ? ? ?|?|k ? = 0

? k

a 1

y1

2 2 2 2 ? ?

D + 2abM a arctan + ib ? +

T, J12 + (a2 + 4)? ? 8

v = (y1 + y2 )?(ib+1/k) ?(?) ? (y1 + y2 )

k

2 y2

2

1

+4 + ib ? = ?|?|k ?

(a ? 0, b ? R)

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

k

Table 3. Reduction by two-dimensional subalgebras of AG2 (1, 2). These ansazes and reduced equations are for equation (11).

Subalgebra Ansatz ? Reduced equation

y2

1 y1 + ay2 ?

v= v

J12 + S + T + 2aM, G1 + P2 exp i ? a arctan t + 1 + ? + (a ? ? 2 )? = ?|?|2 ?

4t t2 + 1

t2 + 1

2

2

t ? 1 y1 + ty2

?(?)

+

(a ? R)

4t t2 + 1

2 2 2

1 y1 + y2 ?

4? ? = ?|?|2 ?

v= v

J12 + 2aM, S + T + 2bM exp i ? b arctan t + ? + 4? + b ? a ? ?

?

t2 + 1 4

t 2+1

2 2

y1 t(y1 + y2 )

?(?)

+ a arctan +

(a ? 0, b ? R)

4(t2 + 1)

y2

Table 4. Reduction by one-dimensional subalgebras of AG(1, 2). These ansazes and reduced equations are for equations (9), (10) and (11).

Subalgebra Ansatz ? Reduced equation

P2 v = ?(?1 , ?2 ) ?1 = t, ?2 = y1 i?1 + ?22 = ?F (|?|)?

2

i

iy2

Solutions of the relativistic nonlinear wave equation

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