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 N-soliton-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.