Page 63 - swp0000.dvi

ˆ
ˆ
where  =  − Ω is phase modulation with wave number (¿ ) and
frequency modulation (Ω ¿ ) respectively. Also,  is the pump carrier
0
wave amplitude, (¿   is the amplitude perturbation). Then equation
0
(3.1) becomes

2
   2
∗
 +  +  | | ( +  )= 0 (3.3)
0
  2
 here is the complex conjugate of . Further, assuming that the ampli-
∗
h ³ ´i
 − Ω 
tude perturbation varies as exp  ˆ  and applying the method
presented in Ref. [144], one can obtain the nonlinear dispersion relation

à !
2
2 | |
0
2
2 ˆ 2
Ω =   ˆ 2  (3.4)
 −

 verify the condition   0 for MI and
All allowed values of ˆ
Ω becomes imaginary. This condition is satisfied for the values of mod-
ˆ
ˆ
ˆ
ulated wave number  which are less than the critical value    =
p
2 | |. In other words, the perturbed wavelengths are larger than
0
ˆ
the critical value 2  (and stable for shorter wavelengths); the maximum
√
ˆ
ˆ
instability growth rate occurs at   =    2 equation (3.4), in the region
ˆ
ˆ
  0 for    experiences to MI the growth rate is obtained as
q
ˆ
ˆ2
Γ =Im Ω()=   ( −  ) (3.5)
2ˆ2 ˆ2

√
ˆ
2
ˆ
with a maximum value Γ  = ||| | for condition  =    2 On the
0
other hand, MI is concerning to the progresses of various kinds of envelope
solitons. In stable wave packets (  0), envelope hole (dark) soliton can
√
be obtained. Consider  = () as a solution of equation (3.1),
where  and  are two real functions to be determined [73,145] as:

 −  1  2
2
 =   ( ) and  = [ +(Ω 0 − )] (3.6)
0
Λ 2 2
50