<<

. 21
( 70 .)



>>

Galilei boosts (19) (they are symmetries of (22)) to these solutions, we ?nd
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



<<

. 21
( 70 .)



>>