# Lax integrability and the peakon problem for the modified Camassa-Holm equation

## TL;DR

This paper investigates the nature of peakon solutions for the modified Camassa-Holm equation, establishing a new weak solution framework, constructing explicit peakon solutions, and analyzing their long-term behavior.

## Contribution

It introduces a distributional compatibility-based weak solution concept, constructs explicit peakon solutions using advanced analysis techniques, and studies their asymptotic dynamics.

## Key findings

- Peakons satisfy the modified Camassa-Holm equation in a new distributional sense.
- Conditions for global existence of peakons are established.
- Large time behavior shows formation of peakon pairs and Toda-like sorting.

## Abstract

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes' continued fractions and multi-point Pad\'{e} approximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons which share asymptotic speeds, as well as Toda-like sorting property.