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

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 $\mathrm{SO}(3)$-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 $G=\mathrm{Diff}(\mathbb{R})$ 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 $G=\mathrm{Diff}(\mathbb{R})$ corresponding to a harmonic map $g:\mathbb{C}\to \mathrm{Diff}(\mathbb{R})$ and find explicit expressions for its peakon-antipeakon solutions, as well.

arXiv:1109.4421 introduced singular solutions of G-strand equations on the diffeos. This paper solves the equations for their pairwise interaction

References in corpus (2)

• Geodesic Flow and Two (Super) Component Analog of the Camassa-Holm Equation
• Dynamics of charged molecular strands