Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves

TL;DR

This paper classifies bifurcations from traveling waves to time-periodic solutions of the Benjamin-Ono equation, using numerical continuation and proving formulas for solutions, revealing complex solution structures and bifurcation behaviors.

Contribution

It introduces a spectrally accurate numerical method to study bifurcation paths and provides exact formulas for time-periodic solutions, advancing understanding of the Benjamin-Ono equation's solution landscape.

Findings

Bifurcation paths either reconnect with different traveling waves or blow up.

Exact formulas for time-periodic solutions involving particle positions and symmetric functions.

Identification of interior bifurcations from existing solution paths.

Abstract

We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We then prove a theorem that gives the mapping from one bifurcation to its counterpart on the other side of the path and exhibits exact formulas for the time-periodic solutions on this path. The Fourier coefficients of these solutions are power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits (circles or epicycles) in the unit disk of the complex plane. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their analytic structure.