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Standing Waves in Near-Parallel Vortex Filaments

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Walter Craig & Chi-Ru Yang

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico, DF, 04510, Mexico

    Carlos García-Azpeitia

  3. Mathematics Department, Shantou University, Shantou, 515063, Guangdong, China

    Chi-Ru Yang

Authors
  1. Walter Craig
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  2. Carlos García-Azpeitia
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  3. Chi-Ru Yang
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Correspondence to Walter Craig.

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Communicated by W. Schlag

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Craig, W., García-Azpeitia, C. & Yang, CR. Standing Waves in Near-Parallel Vortex Filaments. Commun. Math. Phys. 350, 175–203 (2017). https://doi.org/10.1007/s00220-016-2781-x

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