Standing Waves in Near-Parallel Vortex Filaments
| Autor: | Chi-Ru Yang, Carlos García-Azpeitia, Walter Craig |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Physics
Anderson localization Dynamical systems theory 010102 general mathematics Statistical and Nonlinear Physics System of linear equations 01 natural sciences Vortex 010101 applied mathematics Standing wave Cantor set symbols.namesake Classical mechanics symbols 0101 mathematics Hamiltonian (quantum mechanics) Mathematical Physics Bifurcation |
| Zdroj: | Communications in Mathematical Physics. 350:175-203 |
| ISSN: | 1432-0916 0010-3616 |
| DOI: | 10.1007/s00220-016-2781-x |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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