Bäcklund transformations, a simple transformation and exact solutions for dust-acoustic solitary waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions

Abstract

Dusty plasma with inertial dust fluid and two-temperature ions admits both compressive and rarefactive solitary waves. The Korteweg-de Vries equations (KdV-type equations) with cubic nonlinearity at the critical density of low-temperature isothermal ions are considered to discuss properties of dust-acoustic solitary waves. In the vicinity of the critical density of low-temperature ions, a KdV-type equation with mixed nonlinearity is discussed. The method of characteristics is used and the Bäcklund transformations (BTs) are employed to generate new solutions from the old ones. Another new solution of the KdV–mKdV equation is obtained using a simple transformation between the sine-Gordon equation and a linear equation combined with an extension of the tanh method of Malfliet.

Exact Solutions with Jacobi Elliptic Functions of Two Nonlinear Models for Ion-Acoustic Plasma Waves

“…Their dynamics have been studied in a series of papers, one of which gives growth rates of instabilities found numerically. 25) In more realistic situation in which the electrons are non-isothermal, Munro and Parkes 15) showed that, with an appropriate modified form of electron number density proposed by Schamel,21) a reductive perturbation procedure leads to a modified form of the modified ZK equation (20).…”

“…Specific results for dusty plasmas include the possibility that dust acoustic solitary waves admit both compressive and rarefactive solutions. 19,20) However, the KdV equation is considered a spatially one dimensional model. An extensive research work has been done in developing higher dimensional models.…”

Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

“…However, the high-order nonlinear terms must be taken into account in some complicated situations like at the critical density or in the vicinity of the critical velocity [6][7][8]. The modified KdV (mKdV)-typed equation, on the other hand, has recently been discovered, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in the supernova shells and Saturn's F-ring [9].…”

“…The Gardner equation has been investigated in detail and can be used to model such physical situations as the dust-acoustic solitary waves in dusty plasmas [6], internal solitary waves in stable, stratified shear flows in ocean and atmosphere [15], ion acoustic waves in plasmas with a negative ion [16], interfacial solitary waves over slowly varying topographies [17] and wave motion in a nonlinear elastic structural element with large deflection [18].…”

“…1-8.Furthermore, we can explicitly present the N-soliton-like solution of Eq. (4) in the sense of Ref [6]. as below,…”