Painlevé analysis, auto-Bäcklund transformation, and new exact solutions for Schamel and Schamel-Korteweg-de Vries-Burger equations in dust ion-acoustic waves plasma

Abstract

A theoretical investigation of dust-acoustic solitary waves in one-dimensional, collisionless, and unmagnetized dusty plasma consisting of ion fluid, trapped as well as free electrons, and charge fluctuating immobile dust particles is considered. The nonlinear dynamics of dust ion-acoustic waves, whose phase speed is much smaller (larger) than the electron (ion) thermal speed, propagating in such a dusty plasma system is investigated. The reductive perturbation method is employed to reduce the basic set of fluid equations to the Schamel and Schamel-Korteweg-de Vries-Burger (S-KdVB) equations. The Schamel and Schamel-KdVB equations are shown to be non-integrable using Painlevé analysis. The Bäcklund transformations and some new exact solutions are formally derived. Finally, we discussed the results in this paper.

Exact Solutions to a Class of Schamel Nonlinear Equations Modeling Dust Ion-acoustic Waves in Plasma

“…where and are constants. The S-KdVB equation containing a square root nonlinearity describes the nonlinear propagation of ion-acoustic shocks in a dusty plasma with dust charge fluctuations and small deviation from isothermally of electrons [30]. Using the transformation…”

Bilinear Bäcklund transformation, N‐soliton, and infinite conservation laws for Lax–Kadomtsev–Petviashvili and generalized Korteweg–de Vries equations

“…Based on solutions (31), (32), and (33), Figure 4 shows the diffusion of one-soliton solution through solution (31) when t = 3, which maintains its shape except for the phase shift, and the diffusion direction can be changed. Figures 5 and 6 show the diffusion of a two-soliton solution through the solution (32) when t = 3, −3.…”

“…However, this method requires complex calculations and suitable variable transformation. The Bell polynomials play an important role in describing the bilinearizable equations, as they enable us to get the bilinear Bäcklund transformation and Lax pairs for soliton equations 29–53 …”

Bäcklund transformation, auto-Bäcklund transformation and exact solutions for KdV-type equations

“…In the past decades, there has been significant progression in the development of methods such as the inverse scattering method [1][2], Hirotas bilinear method [3], Painlevé expansions [4][5][6][7], truncated Painlevé [8], homogeneous balance method [9][10], the linearized transformation method [11], tanh function method [12][13][14][15]and several ansatz method [16][17]. The Bäcklund transformation (BT) technique is one of the direct methods for generating a new solution of NLEEs from a known solution of that equation (see, for example, [18][19][20]).…”