2014/11/19 by Walter Craig, Craig, W., Carlos García-Azpeitia +3
Computer Science · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1411.5105
openalex publication_date 2014/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.