$G$-Strands and Peakon Collisions on ${\rm Diff}(\mathbb{R})$

TL;DR

This paper investigates the dynamics and collisions of singular solutions called peakons within $G$-strands on the diffeomorphism group of the real line, revealing explicit solutions and connections to classical PDEs.

Contribution

It extends the study of $G$-strands to the infinite-dimensional group ${ m Diff}( eal)$, analyzing peakon collisions and providing explicit solutions for complexified cases.

Findings

Peakons in ${ m Diff}( eal)$ can be explicitly described during collisions.

Solutions reduce to classical PDEs like Laplace and wave equations.

Explicit peakon-antipeakon solutions are derived for complexified systems.

Abstract

A $G$-strand is a map $g:\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. Some $G$-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the ${\rm SO}(3)$-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that $G$-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of $G$-strands when $G={\rm Diff}(\mathbb{R})$ is the group of diffeomorphisms of the real line $\mathbb{R}$, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of $G$-strand equations for $G={\rm Diff}(\mathbb{R})$ corresponding to a harmonic map $g: \mathbb{C}\to{\rm Diff}(\mathbb{R})$ and find explicit expressions for its peakon-antipeakon solutions, as well.