Negative order KdV equation with both solitons and kink wave solutions

TL;DR

This paper introduces an integrable negative order KdV equation capable of supporting both soliton and kink wave solutions, derived from the negative KdV hierarchy and related to the Camassa-Holm equation.

Contribution

It presents the derivation of a new integrable equation with both soliton and kink solutions, including its Lax pair and solution types, expanding understanding of negative order KdV equations.

Findings

Equation admits classical solitons, periodic solitons, and kink solutions.

Lax pair derived confirming integrability.

Equation related to the Camassa-Holm equation via gauge transform.

Abstract

In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions.