A view of the peakon world through the lens of approximation theory

TL;DR

This paper explores the mathematical theory of peakons, special soliton solutions of nonlinear PDEs, highlighting their connections to classical approximation problems and integrable systems.

Contribution

It provides a comprehensive overview linking peakon solutions to approximation theory, distributional Lax pairs, and explicit formulas for peakon dynamics in integrable PDEs.

Findings

Peakons are connected to Padé and Hermite-Padé approximation problems.

Distributional Lax pairs are essential for understanding peakon solutions.

Explicit formulas for peakon solutions are derived through approximation theory.

Abstract

Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with particular emphasis on the connections to classical problems in analysis, such as Pad\'e approximation, mixed Hermite-Pad\'e approximation, multi-point Pad\'e approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising from the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a variety of approximation-theoretic problems, and we illustrate the peakon dynamics with graphics.