Eur. Phys. J. B 53, 343-350 (2006)
https://doi.org/10.1140/epjb/e2006-00378-3

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

¹ Department of Mathematics and LMIB, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China
² Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China
³ CCAST (World Lab.), PO Box 8730, Beijing, 100080, China
⁴ Meteorology Center of Air Force Command Post, Changchun, 130051, China

Corresponding authors: ^a gmwei@buaa.edu.cn - ^b gaoyt@public.bta.net.cn

Received: 30 June 2006
Revised: 6 September 2006
Published online: 18 October 2006

Abstract

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painlevé analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painlevé test. An auto-Bäcklund transformation is provided by use of the truncated Painlevé expansion and symbolic computation. Via the given auto-Bäcklund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.