ñòð. 20 |

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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