Superposition solutions to the extended KdV equation for water surface waves

TL;DR

This paper explores superposition solutions for the extended KdV equation modeling water surface waves, demonstrating that such solutions can be constructed for higher order, non-integrable extensions, challenging the common association between integrability and solution complexity.

Contribution

It shows that superposition solutions can be generated for higher order, non-integrable KdV extensions, expanding the understanding of solution structures beyond integrable cases.

Findings

Superposition solutions exist for extended, non-integrable KdV equations.

No direct link between integrability and the number of wave solutions.

Construction method applicable to various higher order wave equations.

Abstract

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena \cite{KhSa,KhSa14,KhSa15} demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg - de Vries, Nonlinear Schr\"{o}dinger). Here we find that this construction can be repeated for higher order, non-integrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions.