Standing Waves in Near-Parallel Vortex Filaments

Abstract

A model derived in [18] 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.

Standing waves in a counter-rotating vortex filament pair

“…the orbits of the standing waves are orthogonal to the traveling direction of the filaments. While for l 0 = 1, this symmetry implies that In [7] the existence of standing waves for n vortex filaments of equal vorticities from a uniformly rotating central configuration is investigated. In the case of two filaments, the distance w 1 (s, t) satisfies the Schrödinger equation ∂ t w 1 = i ∂ ss w 1 + |w 1 | −2 w 1 , which has the explicit solution ae ia −2 t that corresponds to the solution where the two filaments rotate with frequency a −2 at distance a.…”

“…This article proves that the co-rotating filament pair has families of standing waves with amplitudes varying over a Cantor set for irrational diophantine frequencies a −2 . In order to solve the small divisor problem that appears due the fact that the standing waves have irrational frequencies, [7] implements a Nash-Moser procedure. This result is different but complementary to the existence of standing waves with rational frequencies in the counter-rotating filament pair.…”

Long Time Evolution of Concentrated Euler Flows with Planar Symmetry

“…For other examples of dynamical systems related to fluid mechanics, see for instance [15,17] and the references quoted in the recent paper [7]. Their connection with the fluid physics, proved in some particular cases, is in general an open issue.…”

Braids of the N-body problem I: cabling a body in a central configuration

“…The relevant cases from the physical point of view are the gravitational potential (α = 2) and the logarithmic potential (α = 1). In the latter case, equations (1.1) govern the approximate interaction of N steady vortex filaments in fluids (Euler equation) [25], [4], [26], [10], Bose-Einstein condensates (Gross-Pitaevskii equation) and superconductors (Ginzburg-Landau equation) [13], [9]. Although it is worth mentioning that there are some subtle differences in the equations for steady vortex filaments with respect to equations (1.1.…”