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turbation theory to the basic set of fluid equations leads to a NLS equa-
tion (2.16). In our model, we have assumed that the effect of the force
of gravity is neglected (the dust grain radius, 1)aswellasthere
are no neutrals. F-rings of Saturn are the suitable space plasma obser-
vations that satisfy our conditions: (i) there are no neutrals, (ii) the ra-
tio between the inter-grain distances to plasma Debye radius is less than
one, (iii) the coupling parameter Γ (potential/thermal energy) is less than
one, and (iv) is smaller than 1 . Hence, numerical studies have
been made using plasma parameters close to those values that corre-
spond to Saturn F-rings. The equilibrium electron and dust densities
−3
are 0 =10 0 =10 −3 and dust charge number and mass are
3
12
given as =10 − 10 and = =10 , respectively [120, 139]. The
2
variation of the envelope group velocity, and the frequency, , with pa-
rameters like the charge numbers for charged hot dust and the carrier
wave number is shown in Fig. (2.1). It is clear that the increase of the
negative charge number of hot dust leads to an increase of the envelope
group velocity, , and the frequency, .
In a formal modulational sinusoidal wave, the NLS equation (2.16) has
been derived. Clearly, from Fig. (2.2), for 0 (Red Region), the
propagating carrier wave is modulationaly stable. So, the dark envelope
wave packet may be propagated [89]. On the other hand, the unstable
solution 0 (white region) rogue wave could be created. More
specifically, one of our main motivation is to study the effect of the plasma
parameters like the population of nonthermal ions , carrier wave ,the
equilibrium density of hot dust grains, 0 and the charge numbers for
negatively charged hot dust, on DA rogue wave properties. The rational
solution of NLS equation introduces the rogue wave form, equation (2.19),
as shown in Fig. (2.3). This wave appears suddenly in a small area
39
