Similarity reductions of peakon equations: integrable cubic equations

TL;DR

This paper explores the similarity solutions of integrable peakon equations, linking them to Painlevé equations through transformations, and provides explicit elliptic and algebraic solutions with asymptotic analysis.

Contribution

It establishes connections between peakon equations and Painlevé equations via reciprocal and hodograph transformations, and derives explicit solutions and asymptotic behaviors.

Findings

Similarity reductions relate to Painlevé equations of types II, III, and V.

Explicit elliptic function solutions for traveling waves.

Asymptotic behaviors of similarity solutions are characterized.

Abstract

We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlev\'e type: for the mCH equation, its reduction is of second order and second degree, while for Novikov's equation the reduction is a particular case of Painlev\'e V. Furthermore, we show that each of these two different Painlev\'e-type equations is related to the particular cases of Painlev\'e III that arise from analogous similarity reductions of the Camassa-Holm and the Degasperis-Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlev\'e III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.