Eur. Phys. J. Plus (2020) 135: 113
https://doi.org/10.1140/epjp/s13360-020-00178-1

Regular Article

A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves

Institute of Nonlinear Science, Shaoxing University, Shaoxing, 312000, China

^* e-mail: liuxizhong123@163.com

Received: 10 May 2019
Accepted: 29 October 2019
Published online: 23 January 2020

Abstract

A nonlocal two-layer fluid model is constructed through a simple symmetry reduction from the local one. Then, a general variable coefficients nonlocal KdV (VCKdV) equation with shifted space parity and delayed time reversal is derived from it by using multi-scale expansion method, with and without the so-called average method, respectively. A non-auto-Bäcklund transformation between the VCKdV equation and a constant coefficients KdV (CCKdV) equation is established. By using this transformation, various exact solutions of the VCKdV equation can be obtained from the seed solutions of the CCKdV equation. As some concrete examples, one solitary wave solution and two kinds of periodic wave solutions are given. Due to the inclusion of arbitrary functions in these solutions, they possess abundant dynamical behaviors with some of them analyzed graphically.