Page 50 - swp0000.dvi

4
4
3
      2
 4 =  0 5 
2
2
2(    0 − 3 0  )
¢
¡ 2 2 2 2 2
 5 =6∆     0 − 3 0  −    0  (5    0
3 0  2
2 2 2
+21 0  ) − [3   0  (    0 +
2
    0
¡
¢
2
2
 0  ) − 2∆     0 − 3 0  2 2 ]
µ ¶ 2 2 µ ¶
(20)  0  0   2 (20) (20)

 6 =  − −  0 +  
3 2 2 2 2 1
       
2
2
     2 ∙ (20) 1 ¡ ¢ (20) ¸
2
+  0 −2 0  −     0 +3 0  2  
2 2
2
(    0 − 3 0  ) 2    0 1
µ ¶
2
  0  0
 7 = +(1+3) 
2   3   3
A negative sign for the product of the coefficients  is required for
amplitude modulation stability. For positive  the amplitude grows
and large wave is created. Many analytical methods have been used to
find rouge wave solutions from NLS equation [88, 106, 109]. The rational
solution is the important one that is given by [105]
s
∙ ¸
2 4(1 + 4)
 = − 1 exp(2)  (2.19)
2
 1+16  +4 2
2
(1)
For simplicity, we have set  ≡  The solution, equation (2.19), refers
1
to an amount of wave energy concentrated in a relatively very small area
in space. The rogue wave is usually an envelope of a carrier wave with a

wavelength smaller than the central region of the envelope.

2.4 Results and Discussion

In this chapter, the nonlinear DAWs in an unmagnetized dusty plasma

having Boltzmann electrons, nonthermal ions, hot and cold dust grains

have been investigated. The application of the derivative expansion per-

38