A family of wave equations with some remarkable properties

TL;DR

This paper introduces a family of nonlinear dispersive wave equations with special properties, including integrability, conservation laws, and explicit solutions like kinks and peakons, by linking them to the KdV equation.

Contribution

It identifies integrable cases within the family, constructs Lax pairs and transformations, and derives explicit solutions, advancing understanding of nonlinear dispersive wave equations.

Findings

Existence of square integrable solutions for the family.

Complete integrability of one member via Lax pair.

Explicit kink and peakon solutions derived from KdV.

Abstract

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found to two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair we construct a Miura-type transformation relating the original equation to the KdV equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schr\"odinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation.