Peakons

TL;DR

This paper reviews peakon solutions, which are singular solutions of the dispersionless Camassa-Holm equation and its generalization EPDiff, highlighting their mathematical structure and Hamiltonian dynamics.

Contribution

It provides a comprehensive review of peakon solutions within the context of asymptotic expansions and variational principles, emphasizing their Hamiltonian structure and momentum map formulation.

Findings

Peakons are singular solutions of the dispersionless CH and EPDiff equations.

Peakons can be understood as momentum maps leading to canonical Hamiltonian dynamics.

The reduction of peakon solutions reveals their Poisson geometric structure.

Abstract

The peakons discussed here are singular solutions of the dispersionless Camassa-Holm (CH) shallow water wave equation in one spatial dimension. These are reviewed in the context of asymptotic expansions and Euler-Poincar\'e variational principles. The dispersionless CH equation generalizes to the EPDiff equation, whose singular solutions are peakon wave fronts in higher dimensions. The reduction of these singular solutions of CH and EPDiff to canonical Hamiltonian dynamics on lower dimensional sets may be understood, by realizing that their solution ansatz is a momentum map, and momentum maps are Poisson.