A new mathematical approach for finding the solitary waves in dusty plasma

Abstract

In a sequel to a recent work [Das, Sarma, and Talukdar, Phys. Plasmas 5, 63 (1998)], the different nonlinear plasma-acoustic waves, based on the fluid approximation, have been derived showing the coexistences of dust-acoustic waves in plasmas contaminated by dust-charged grains. The features of the nonlinear waves, depending on the plasma composition, describe various natures of solitary waves. A new formalism, known as the tanh method and stemming from the modified simple wave solution technique, has been developed for finding the soliton propagation in the nonlinear plasma wave dynamics. The method is straightforward, with minimal mathematical manipulation, finding the heuristic formation and propagation of ion-acoustic solitary waves in the dusty plasma. The main aim is, based on the tanh method, to revisit the results in a simpler case and extending them to explain the behavior of higher-order nonlinear waves derived in generalized multicomponent plasmas. The theoretical observations highlight the salient features of nonlinear waves coexisting with the dust-acoustic wave. The new findings might expect the effect, because of the dust-charged grains, to be the common feature of nonlinear waves in the dusty plasmas, and could be of interest for future experiments in laboratory as well as in space plasmas.

Comment on “A new mathematical approach for finding the solitary waves in dusty plasma” [Phys. Plasmas 5, 3918 (1998)]

“…1 The reason is fairly obvious and has been discussed in many papers. Essentially, all counter arguments are based on the structure of a nonlinear evolution equation as a proper balance between slow time changes, and nonlinear and wave dispersive effects.…”

“…Of course, one can find single solitary wave solutions for equations that are not completely integrable like the Schamel equation 12,13 and extensions of it discussed by Das and Sarma. 1 ͑4͒ Finally, it is equally misleading to claim that dusty plasmas were included among the possible applications of the early literature on nonlinear waves in multicomponent plasmas, as in the paper by Das and Tagare. 14 Although, with hindsight, we can now apply the formalisms developed in these papers to waves in dusty plasmas with constant dust charges, one cannot say that these papers predicted dusty plasma modes, because in these papers there is no reference whatsoever to dusty plasmas.…”

N-soliton solutions and asymptotic analysis for a Kadomtsev–Petviashvili–Schrödinger system for water waves

“…With the development of soliton theory, the Korteweg-de Vries (KdV) model has been employed to describe such nonlinear phenomenon as in plasmas, shallow water waves, fluid dynamics, and lattice vibrations of a crystal at low temperatures [1][2][3][4]. With the higher dispersion, the fifth-order KdV equation has been used in solid-state physics, fluid physics, plasma physics, and quantum field theory [5][6][7][8][9].…”

Vadermonde-Type Odd-Soliton Solutions for the Whitham–Broer–Kaup Model in the Shallow Water Small-Amplitude Regime

“…Seeking for the soliton solutions of the nonlinear evolution equations (NLEEs) is of importance since such equations can describe the diverse physical aspects [1][2][3][4][5][6][7][8][9][10][11][12]. Darboux transformations (DTs) based on the Lax pair are a method to get the soliton solutions of some NLEEs from the seeds .…”

“…Especially, the N -fold DT, which can be interpreted as the superposition of a single DT, has been applied to certain NLEEs for deriving the multi-soliton solutions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Advantage of the N -fold DT is that the problem solving of a NLEE is finally reduced to solve a linear system, which enables us to generate the multi-soliton solutions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] with symbolic computation [1][2][3][4][5][6][7][8][9][10][11][12].…”