Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

Contreras, Andres; Jerrard, Robert L.

doi:10.1007/s00039-017-0425-8

Cited by 21 publications

(33 citation statements)

References 36 publications

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“…In this work only degree d i = +1 were considered. This follows an earlier work on the interaction of vortex filaments for the Ginzburg-Landau equation by Contreras-Jerrard [14]. The result in [26] is based on variational arguments, and therefore only finite energy solutions are considered in cylindrical domains of the form ω × R where ω ⊂ R 2 is bounded, with periodicity in the third variable.…”

Section: Introductionmentioning

confidence: 85%

Interacting helical traveling waves for the Gross-Pitaevskii equation

Dávila¹,

Pino²,

Medina³

et al. 2021

Preprint

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We consider the 3D Gross-Pitaevskii equation i∂tψ + ∆ψ + (1 − |ψ| 2 )ψ = 0 for ψ : R × R 3 → C and construct traveling waves solutions to this equation. These are solutions of the form ψ(t, x) = u(x 1 , x 2 , x 3 − Ct) with a velocity C of order ε| log ε| for a small parameter ε > 0. We build two different types of solutions. For the first type, the functions u have a zero-set (vortex set) close to an union of n helices for n ≥ 2 and near these helices u has degree 1. For the second type, the functions u have a vortex filament of degree −1 near the vertical axis e 3 and n ≥ 4 vortex filaments of degree +1 near helices whose axis is e 3 . In both cases the helices are at a distance of order 1/(ε | log ε|) from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.

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Section: Introductionmentioning

confidence: 85%

Interacting helical traveling waves for the Gross-Pitaevskii equation

Dávila¹,

Pino²,

Medina³

et al. 2021

Preprint

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“…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.…”

Section: Model: the N -Body And N -Vortex Filament Problemsmentioning

confidence: 99%

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

Fontaine

García-Azpeitia

2021

Nonlinearity

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We prove the existence of periodic solutions of the N = (n + 1)-body problem starting with n bodies whose reduced motion is close to a non-degenerate central configuration and replacing one of them by the center of mass of a pair of bodies rotating uniformly. When the motion takes place in the standard Euclidean plane, these solutions are a special type of braid solutions obtained numerically by C. Moore. The proof uses blow-up techniques to separate the problem into the n-body problem, the Kepler problem, and a coupling which is small if the distance of the pair is small. The formulation is variational and the result is obtained by applying a Lyapunov-Schmidt reduction and by using the equivariant Lyusternik-Schnirelmann category.

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“…The perturbation from straight filaments oscillates approximately as u(s, t) ∼ r(e −it ) cos s. confinement, the core of a higher index vortex filament separates into a fine structure described by the system of near-parallel index-1 filaments, whose positions are described by the system (1). This asymptotic description has been rigorously established in [6] in the stationary case, and in the case of nontrivial time dependent evolution in [16]. To our knowledge, the rigorous analytic justification of (1) as a model of vortex filaments for the Euler equations of fluid dynamics is open.…”

Section: Introductionmentioning

confidence: 95%

Standing Waves in Near-Parallel Vortex Filaments

Craig

García-Azpeitia

Yang

2016

Commun. Math. Phys.

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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.

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Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

Cited by 21 publications

References 36 publications

Interacting helical traveling waves for the Gross-Pitaevskii equation

Interacting helical traveling waves for the Gross-Pitaevskii equation

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

Standing Waves in Near-Parallel Vortex Filaments

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