Standing Waves in Near-Parallel Vortex Filaments

A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995 ) 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 a...

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Veröffentlicht in:Communications in mathematical physics 2017-02, Vol.350 (1), p.175-203
Hauptverfasser: Craig, Walter, García-Azpeitia, Carlos, Yang, Chi-Ru
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description A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995 ) 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.
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subjects Anderson localization
Bifurcations
Circulation
Classical and Quantum Gravitation
Complex Systems
Computational fluid dynamics
Dimensional analysis
Filaments
Fine structure
Mathematical and Computational Physics
Mathematical models
Mathematical Physics
Oscillators
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Standing waves
Superconductors
Theoretical
Vortex filaments
Vortices
title Standing Waves in Near-Parallel Vortex Filaments
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