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On writing u = ???n/2 , this integral reduces to
du n
= ? (±? + C1 ).
2
? au2
2?
n+2

For ? > 0, a < 0 we obtain (after some calculation)
v
?
1 ? cosh n ?a(C1 ± ?)
u2 =
a(n + 2)
or

a(n + 2) 1
v
n
?= .
1 ? cosh n ?a(C1 ± ?)
?

Finally, noting that we have ? = y1 = ?x, in the notation of Section 2.3, we ?nd that

a(n + 2) 1
? = e?i(a(?x)+(?x)/2) v
n

1 ? cosh n ?a(C1 ± ?x)
?

is a solution of

2? = ??|?|n ?,
88 P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

when ? > 0, a < 0. If we take ? > 0, a > 0, then we obtain, with similar calculations,
that

a(n + 2) 1
? = e?i(a(?x)+(?x)/2) v
n

1 ? cos n a(C1 ± ?x)
?

is a solution of

2? = ??|?|n ?.

Case 2. A = 0, n = 2, a = 0. In this case we also have ? = const, and (15) becomes
1
?2 + a?2 ? ??4 = C. (16)
?
2
Equation (16) can be solved using Jacobian elliptic functions. For the de?nitions, we
refer to [29]. Following [30], we take a, ? and C as functions of a real parameter ?,
with |?| < 1, and using the generic notation E(?, ?) for solutions of (16), we have the
following table of exact solutions:

E(?, ?) a(?) ?(?) C(?)
1 + ?2 2?2
sn 1
1 ? 2?2 ?2?2 1 ? ?2
cn
?2 ?2 ? 1
?2
dn
1 + ?2 ?2
ns = 1/sn 2
1 ? 2?2 2(1 ? ?2 ) ??2
nc = 1/cn
?2 ? 2 2(?2 ? 1) ?1
nd =1/dn
?2 ? 2 2(1 ? ?2 )
sc = sn/cn 1
1 ? 2?2 2?2 (?2 ? 1)
sd = sn/dn 1
?2 ? 2 1 ? ?2
cs = cn/sn 2
1 + ?2 2?2
cd = cn/dn 1
1 ? 2?2 ?2 (?2 ? 1)
ds = dn/sn 2
1 + ?2 ?2
dc = dn/cn 2

Using this table and the notation of Section 2.3, we ?nd that

? = e?i(a(?)(?x)+(?x)/2) E(?x, ?)

is an exact solution of

2? = ??(?)|?|2 ?,

where a(?) and ?(?) are the appropriate functions of the parameter ?, as given in the
above table. This gives us elliptic solutions of a nonlinear relativistic wave equation.
We note that solutions of nonlinear wave equations in terms of elliptic functions were
obtained by Petiau [35]. The solutions we present here are for a di?erent nonlinearity.
Case 3. n = 2, a = 0. If we put n = 2 and a = 0 in (15), we obtain the equation

A2 ?
? + 2 = ?4 + C.
2
?
? 2
Solutions of the relativistic nonlinear wave equation 89

On multiplying this equation by ?2 , and putting z = ?2 , we obtain the following
equation for z:
8A2
? 8C
z?
z2 = 4z 3 +
? ,
2 ? ?
which gives us the solution

1
z=? ?? ,
2

where ?(?) is the Weierstrass elliptic function (see [29]), provided that 27A4 +8C 3 /? =
0 (the equation (d?/ds)2 = 4? 3 ?g2 ? ?g3 has ?(s) as solution provided g2 ?27g3 = 0).
3

From this it is straightforward to deduce that
v?
2 (?x)
1 ?x 2A d?
?(?x) exp ?
?= ? +
2 2 ? ?(?)

is a solution of
2? = ??|?|2 ?.
Next we turn to the case G1 + aP1 , G2 in Table 1. The reduced equation is

?1 1 1
? = ?i?F (|?|)?.
?+ +
??a ?
2
Using the amplitude-phase representation ? = ?ei? in this equation, as before, we ?nd
the following system:
1 1 1
(17)
?+
? + ? = 0,
??a ?
2
?
? = ??F (?). (18)

Equation (17) integrates immediately to give
C
?= ,
?(? ? a)
where C is a constant of integration. Using this, (18) now yields

C
? = ?? F d? + C1 .
?(? ? a)
Combining this with the corresponding ansatz for the solution v of (8), and using the
notation of Section 2.3, we obtain that
C
?
?=
(?x)2 ? a(?x)
?x
?x (?x)2 + (?x)2
C
? exp ?i ? F d? + +
?(? ? a) 2 4?x
90 P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk

is an exact solution of

2? = ??F (|?|)?,

and when F (?) = ? n , with n ? 2, we have
C
?
?=
(?x)2 ? a(?x)
Cn ?x (?x)2 + (?x)2
? exp ?i ?? + +
(n ? 1)[(?x)2 ? a(?x)](n?1)/2 2 4?x
as an exact solution.

4 Special solutions of some nonlinear
complex wave equations
In this section we give some particular solutions of some multi-dimensional hyperbolic
(‘relativistic’) equations which can be reduced to Schr?dinger equations with our
o
ansatz (7). In some cases, the nonlinear Schr?dinger equation involved admits a soliton
o
solution in 1+1 space-time.
First we take the hyperbolic equation

2? = ?|?|n ?.

The ansatz (7) (with k = ?1/2) reduces this to

ivt + ?v + ?|v|n v = 0,

as we have already noted. It is a simple matter to verify that for ? = a2 b2 n
2 2
+1
n
we have
exp(4ia2 b2 t/n2 )
v=
cosh2/n (ba · y)
as a solution. Here a = (a1 , a2 ), y = (y1 , y2 ), where a = (a1 , a2 ) is an arbitrary vector
and b an arbitrary real number. Applying the Galilean boosts (which are symmetries
of the above nonlinear Schr?dinger equation)
o
1
Ga = t?a + ixa (v?v ? v ?v ) (19)
??
2
(where a = 1, 2) to this solution, we obtain the solution

exp[i(4a2 b2 t/n2 + V · y/2 ? V 2 t/4)]
v= ,
cosh2/n (ba · (y ? V t))
where V = (V1 , V2 ) is an arbitrary vector. For n = 2 and in 1+1 space-time, we have

exp[i(a2 b2 t + V y/2 ? V 2 t/4)]
v= ,
cosh(ab(y ? V t))
Solutions of the relativistic nonlinear wave equation 91

which is the Zakharov–Shabat soliton. Finally, using (7), we obtain
exp[i(??x/2 + 4a2 b2 (?x)/n2 + (V1 (?x) + V2 (?x))/2 ? V 2 (?x)/4)]
?=
cosh2/n (b[a1 (?x ? V1 t) + a2 (?x ? V2 (?x))])
as a solution of
2 2
2? = a2 b2 + 1 |?|n ?
n n
in 1+3 space-time.
There are some other hyperbolic equations which can be reduced to nonlinear
Schr?dinger equations, but with nonlinearities involving derivatives. The hyperbolic
o
equations of the form
2? = ?F (|?|, |?|µ |?|µ )? (20)
can also be reduced to nonlinear Schr?dinger equations with derivative nonlinearities,
o
using the same ansatz (7) (which is not surprising as the same symmetry operator is
responsible for the ansatz). Indeed, ansatz (7) with k = ?1/2 gives us
ivt + ?v + ?F (|v|, ?|v|a |v|a )v = 0, (21)
where |v|a |v|a = |v|21 + |v|22 . Equations of the type (21) were discussed in [21] from
y y
a group-theoretical point of view. One of this type of Schr?dinger equations is
o
|v|a |v|a
(22)
ivt + ?v = 2 v,
|v|2
|v|a |v|a
with ? = ?2 and F (|v|, |v|a |v|a ) = |v|2 . Equation (22) admits the two solutions:

exp(?ia2 t) exp(?ia2 t)
v=A , v=A ,
cosh(a · y) sinh(a · y)
where a = (a1 , a2 ) is an arbitrary vector and A is an arbitrary number. Applying the
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