An infinite branching hierarchy of time-periodic solutions of the Benjamin-Ono equation

TL;DR

This paper introduces a unified, explicit representation of time- and space-periodic solutions to the Benjamin-Ono equation, revealing an infinite hierarchy of bifurcations and particle dynamics.

Contribution

It provides a novel, comprehensive framework for understanding periodic solutions and bifurcations in the Benjamin-Ono equation, including explicit Fourier and particle trajectory descriptions.

Findings

Unified representation of solutions with fixed spatial and varying temporal periods

Explicit description of Fourier mode evolution and particle trajectories

Identification of new bifurcation mechanisms including degenerate cases

Abstract

We present a new representation of solutions of the Benjamin-Ono equation that are periodic in space and time. Up to an additive constant and a Galilean transformation, each of these solutions is a previously known, multi-periodic solution; however, the new representation unifies the subset of such solutions with a fixed spatial period and a continuously varying temporal period into a single network of smooth manifolds connected together by an infinite hierarchy of bifurcations. Our representation explicitly describes the evolution of the Fourier modes of the solution as well as the particle trajectories in a meromorphic representation of these solutions; therefore, we have also solved the problem of finding periodic solutions of the ordinary differential equation governing these particles, including a description of a bifurcation mechanism for adding or removing particles without destroying periodicity. We illustrate the types of bifurcation that occur with several examples, including degenerate bifurcations not predicted by linearization about traveling waves.