Topological Bifurcations and Reconstruction of Travelling Waves

TL;DR

This paper investigates the exact reconstruction of periodic travelling waves in Lie-Poisson equations on the Virasoro group, revealing integrability and bifurcations in particle motion related to topological changes in orbits.

Contribution

It demonstrates the exact reconstruction of travelling wave solutions on the Virasoro group for arbitrary Hamiltonians and identifies topological bifurcations in particle drift for the Camassa-Holm equation.

Findings

Exact reconstruction of solutions regardless of Hamiltonian

Expression of particle drift in terms of orbit parameters

Identification of orbital bifurcations with topological changes

Abstract

This paper is devoted to periodic travelling waves solving Lie-Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which need not be quadratic), provided the wave belongs to the coadjoint orbit of a uniform profile. Equivalently, the corresponding "fluid particle motion" is integrable. Applying this result to the Camassa-Holm equation, we express the drift of particles in terms of parameters labelling periodic peakons and exhibit orbital bifurcations: points in parameter space where the drift velocity varies discontinuously, reflecting a sudden change in the topology of Virasoro orbits.