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arXiv · 1411.5105

Standing waves in near-parallel vortex filaments

W. Craig·C. Garcia-Azpeitia·C-R. Yang
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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.

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W. Craig, C. Garcia-Azpeitia, C-R. Yang. 2016-08-19. Standing waves in near-parallel vortex filaments. https://arxiv.org/abs/1411.5105

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