Characterization of Rational Solutions of a KdV-Like Equation

TL;DR

This paper characterizes rational solutions of a KdV-like equation, showing limitations on polynomial degrees and confirming a conjecture about these solutions, advancing understanding of their structure.

Contribution

It provides a complete characterization of rational solutions for a KdV-like equation and proves a conjecture regarding their polynomial degree constraints.

Findings

No solutions with degree greater than 5 in the spatial variable.

Confirmed the conjecture of Zhang and Ma about these solutions.

Identified a class of polynomials satisfying a quadratic difference equation.

Abstract

We characterize the rational solutions to a KdV-like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation. We use a particular class of polynomials satisfying a quadratic difference equation to obtain that there are no solutions of the bilinear equation with degree (in the spatial variable) greater than 5. As a byproduct, we answer positively a conjecture of Yi Zhang and Wen-Xiu Ma about these solutions.