Similarity reductions of peakon equations: the $b$-family

TL;DR

This paper investigates the $b$-family of shallow water wave equations, focusing on peakon solutions and their similarity reductions, revealing integrability properties for specific parameter values and non-Painlevé behavior otherwise.

Contribution

It analyzes similarity reductions of the $b$-family, connecting special cases to Painlevé III and demonstrating non-Painlevé equations for other parameters.

Findings

For $b=2,3$, similarity reductions relate to Painlevé III.

Other $b$ values lead to non-Painlevé type equations.

Peakons are key solutions in the $b$-family.

Abstract

The $b$-family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases $b=2,3$ (the Camassa-Holm and Degasperis-Procesi equations, respectively) the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter $b$ it is non-integrable. After a discussion of travelling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the $b$-family, and show that when $b=2$ or $3$ this similarity reduction is related by a hodograph transformation to particular cases of the Painlev\'e III equation, while for all other choices of $b$ the resulting ordinary differential equation is not of Painlev\'e type.