Page 36 - swp0000.dvi
P. 36
1.9.4 Solution for Rogue Wave
The standard form of NLS equation is (1.11); namely
2
2
+ + || =0 (1.16)
2
where is a temporal variable, and is the spatial variable in the moving
frame.
Equation (1.16) has a general rational solution in the form [107],
s
µ ¶
2 ( )+ ( )
( )= (−1) + (2) (1.17)
( )
where ( ), ( ) and ( ) are polynomials in and ,and
( ) has no real zeros. Index is the order of solution. In lowest-order
solution =1, one gets [88-108]
s
∙ ¸
2 4(1 + 4)
( )= − 1 2 (1.18)
1
1+16 +4 2
2
2
In second-order solution with =2, the polynomials 2 ( ), 2 ( )
and 2 ( ) in this solution are given by [107]
√
¢
2
¡
4
2 ( )= 36 − 48 − 144 2 4() +1 − 24 2 1 − 960 () 4
2
−864 () +48 2 () (1.19)
h ³ √ ´ ¡ ¢
2
4
2
2 ( )= 24 15 − 4 +12 +2 2 1 − 8() 3 2 +1 − 16 () 5
µ ¶¸
1
2 2
+ 2 2() − − (1.20)
2
25
