Stability of closed solutions to the VFE hierarchy with application to the Hirota equation

TL;DR

This paper develops a framework linking the stability of closed solutions in the VFE hierarchy to the NLS hierarchy, applying it to Hirota equation solutions and discussing solitons.

Contribution

It introduces a general method to analyze stability of VFE hierarchy solutions via the NLS hierarchy and Floquet spectrum, with applications to the Hirota equation.

Findings

Established a connection between AKNS Floquet spectrum and solution stability.

Applied the framework to Hirota equation solutions.

Discussed the applicability to soliton solutions.

Abstract

The Vortex Filament Equation (VFE) is part of an integrable hierarchy of filament equations. Several equations part of this hierarchy have been derived to describe vortex filaments in various situations. Inspired by these results, we develop a general framework for studying the existence and the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between the VFE and the nonlinear Schr\"odinger (NLS) hierarchies. Our results show that it is possible to establish a connection between the AKNS Floquet spectrum and the stability properties of the solutions of the filament equations. We apply our machinery to solutions of the filament equation associated to the Hirota equation. We also discuss how our framework applies to soliton solutions.