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and
¡ 2 ¢ ¡ 2 2 6
¢
6
2 ( )= 8 +12 4 4() +1 +6 2 3 − 4() +64 () +9
h √ i
4 2 ¡ 2 2 ¢
+432 () +396 () + 1 1 +2 2 12 () − 2 +3
£ ¡ 2 ¢¤
2
+ 2 2 +4 () 6 − 4() − 9 (1.21)
where the free parameters 1and 2 describe the relative positions of three
first-orders breathers in triplets.
1.10 Previous Works
No doubt, the study of plasmas with charged dust particles are very im-
portant for many environmental systems in astrophysics [49]. The charge
on dust grain introduces a number of electrostatic dust wave modes [67-
71]. The astrophysical dusty plasmas waves can be excited such as, dust
acoustic (DA) mode, DIAs and dust-lattice (DL) waves [67-71,109]. More
specifically, the grain charging occurs due to different processes [110, 111].
El-Labany et al. [112] studied the three-dimensional stable and unstable
DAWs in opposite polarity dusts with dust size distribution. The proper-
ties of nonlinear DAWs in two-temperatures dusty plasma have been in-
vestigated by many authors [113-115]. Cairns et al. [44] supposed electron
nonthermality to examine the propagation of nonlinear acoustic solitary
waves observed by FREJA satellite. In astrophysical space plasmas, ob-
servations indicated the presence of nonthermal ion populations in Earth’s
bow shock and around the Earths foreshock [115-117]. Also, ASPERA
observed nonthermal ions flux from the upper ionosphere of Mars [118].
Fast nonthermal ions have been recently observed by the Nozomi satellite
in the vicinity of the Moon [119]. El Wakil et al. [120] examined the effect
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