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Spectral stability of solutions to the vortex filament hierarchy

Banff International Research Station for Mathematical Innovation and Discovery

The Vortex Filament Equation (VFE) is part of an integrable hierarchy of filament equations. Several equations in 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 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.

Mathematics; Partial differential equations; Dynamical systems and ergodic theory; Functional analysis

video/mp4

Author affiliation: College of Charleston

2017-12-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64048

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/