Standing waves in near-parallel vortex filaments

TL;DR

This paper constructs families of standing wave solutions in a vortex filament model relevant to fluid dynamics and Bose-Einstein condensates, using advanced mathematical techniques to handle the complex bifurcation problem.

Contribution

It introduces a novel application of Nash-Moser and Lyapunov-Schmidt methods to construct standing waves in a vortex filament model with Hamiltonian PDEs.

Findings

Existence of co-rotating vortex filament solutions

Application of Nash-Moser method to bifurcation analysis

Solutions form a Cantor set of parameters

Abstract

A model derived in [14] for n near-parallel vortex filaments in a three dimensional fluid region takes into consideration the pairwise interaction between the filaments along with an approximation for motion by self-induction. The same system of equations appears in descriptions of the fine structure of vortex filaments in the Gross -- Pitaevski model of Bose -- Einstein condensates. In this paper we construct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a central configuration. This result applies to any pair of vortex filaments with the same circulation, corresponding to the case n=2. The model equations can be formulated as a system of Hamiltonian PDEs, and the construction of standing waves is a small divisor problem. The methods are a combination of the analysis of infinite dimensional Hamiltonian dynamical systems and linear theory related to Anderson localization. The main technique of the construction is the Nash-Moser method applied to a Lyapunov-Schmidt reduction, giving rise to a bifurcation equation over a Cantor set of parameters.