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2
where  is the speed of the pulse that oscillates with frequency Ω 0 (=  | | )
0
q
at rest. The width Λ is given by Λ = 2 . On the contrariwise, for   0,
 0
stable wave packet dark envelope soliton will be produced in the form:

 −  1  2
2
 =   ( )and  = [ − ( − 2  )] (3.7)
1
1
Λ 1 2 2
which is related to a localized hole traveling with speed  . The width of
q
¯ ¯
the pulse is Λ 1 = 2( ¯  ¯ ) 
 1

3.3 Results and Discussion

We investigated the nonlinear dust-acoustic MI in an unmagnetized dusty
plasma having electrons, nonthermal ions, hot and cold dust grains using

NLS equation (3.1). In our model, numerical studies have been made using

plasma parameters close to those values correspond to Saturn F-rings. The

−3
equilibrium electron and dust densities are  0 =10   0 =10  −3 and
dust charge and mass are given as  =10 − 10 and   =   =10  ,
2
12
3

respectively [121,140]. The variation of the coefficients of dispersion and
nonlinear terms  and  with the population of nonthermal ions ,the

carrier wave number , the equilibrium density of hot dust grains  0 and
the charge number for negatively charged hot dust,   are shown in Figs.

(3.1- 3.10). It is clear from Figs. (3.1-3.5) that increasing the population

of nonthermal ions  and negative charge number of hot dust   leads
to an increase of the envelope soliton width. Also, Figs.(3.6-3.10) show

ˆ
the variation of  with  for different values of plasma parameters. It is
noted that, there are critical values for carrier wave number. Moreover,
ˆ
the ratio () versus  for different plasma variables is shown in Figs.
(3.10-3.14). For  =0 the ratio ()−  ∞ the corresponding values

ˆ
of wave number (=   ) are called critical wave numbers for the MI. It is

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