Existence of polynomial solutions for an extended bilinear equation of Korteweg-de Vries equation

Abstract

In this paper, we investigate the properties of polynomial solutions to an extended bilinear equation of Korteweg-de Vries (KdV) equation by means of differential field and algebraic extension, and prove that the degrees of its polynomial solutions are less than 5, then all rational solutions for the KdV-like equation are provided. Furthermore we point out that some obtained properties can be extended to other bilinear equations and can simplify the computation of finding rational solutions of KdV equation family. With symbolic computation, the four classes nontrivial rational solutions to KdV equation are presented.