The Painlevé property, Bäcklund transformation, Lax pair and new analytic solutions of a generalized variable-coefficient KdV equation from fluids and plasmas

Abstract

A generalized variable-coefficient Korteweg-de Vries (KdV) equation with variable-coefficients of x and t from fluids and plasmas is investigated in this paper. The explicit Painlevé-integrable conditions are given out by Painlevé test, and an auto-Bäcklund transformation is presented via the truncated Painlevé expansion. Under the integrable condition and auto-Bäcklund transformation, the analytic solutions are provided, including the soliton-like, periodic and rational solutions. Lax pair, Riccati-type auto-Bäcklund transformation (R-BT) and Wahlquist-Estabrook-type auto-Bäcklund transformation (WE-BT) are constructed in extended AKNS system. One-soliton-like and two-soliton-like solutions are obtained by R-BT and nonlinear superposition formula is obtained by WE-BT. The bilinear form and Nsoliton-like solutions are presented by Bell-polynomial approach. Based on the obtained analytic solutions, the propagation characteristics of waves effected by the variable coefficients are discussed.

Lie symmetry analysis of a variable coefficient Calogero–Degasperis equation

“…Although, to our knowledge, the problem of group classification for variable coefficient KdV equation was considered in a number of articles, see for example in [26,27] and for more general equations, see for example in recent publications [34][35][36][37][38], the corresponding problem for the class (9) does not appear in the literature. We present the results of Lie group classification of (9) in the appendix B.…”

Oceanic Shallow-Water Investigations on a Variable-Coefficient Davey–Stewartson System

“…where the subscripts denote the partial derivatives, x is the scaled horizontal coordinate, y denotes the scaled space coordinate perpendicular to x, t is the scaled time, the complex function u(x, y, t) is the amplitude of a surface wave packet, the real function v(x, y, t) is the velocity potential of the mean flow interacting with the surface wave, the parameter ε = 1 characterizes the focusing case, and the parameter ε = −1 characterizes the defocusing case [24]. When the media are inhomogeneous or the boundaries are nonuniform, variablecoefficient models are able to describe various situations more realistically than their constant-coefficient counterparts [26]. In this paper, we focus our interest on the variablecoefficient Davey-Stewartson (vcDS) equation for ocean waves, ultra-relativistic degenerate dense plasmas, and Bose-Einstein condensates [24]:…”