Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel; G. Orlandi; Didier Smets

Journal of the European Mathematical Society (2004)

  • Volume: 006, Issue: 1, page 17-94
  • ISSN: 1435-9855

Abstract

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We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N 3 . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3 ).

How to cite

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Bethuel, Fabrice, Orlandi, G., and Smets, Didier. "Vortex rings for the Gross-Pitaevskii equation." Journal of the European Mathematical Society 006.1 (2004): 17-94. <http://eudml.org/doc/277427>.

@article{Bethuel2004,
abstract = {We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N\ge 3$. We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if $N=3$).},
author = {Bethuel, Fabrice, Orlandi, G., Smets, Didier},
journal = {Journal of the European Mathematical Society},
keywords = {NLS; Gross-Pitaevskii; Ginzburg-Landau; Vortex rings; finite energy solutions},
language = {eng},
number = {1},
pages = {17-94},
publisher = {European Mathematical Society Publishing House},
title = {Vortex rings for the Gross-Pitaevskii equation},
url = {http://eudml.org/doc/277427},
volume = {006},
year = {2004},
}

TY - JOUR
AU - Bethuel, Fabrice
AU - Orlandi, G.
AU - Smets, Didier
TI - Vortex rings for the Gross-Pitaevskii equation
JO - Journal of the European Mathematical Society
PY - 2004
PB - European Mathematical Society Publishing House
VL - 006
IS - 1
SP - 17
EP - 94
AB - We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N\ge 3$. We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if $N=3$).
LA - eng
KW - NLS; Gross-Pitaevskii; Ginzburg-Landau; Vortex rings; finite energy solutions
UR - http://eudml.org/doc/277427
ER -

Citations in EuDML Documents

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  1. F. Bethuel, G. Orlandi, D. Smets, Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics
  2. Philippe Gravejat, Decay for travelling waves in the Gross–Pitaevskii equation
  3. Mihai Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems
  4. Fabrice Béthuel, Philippe Gravejat, Jean-Claude Saut, Ondes progressives pour l’équation de Gross-Pitaevskii
  5. Didier Smets, Problèmes d’évolution liés à l’énergie de Ginzburg-Landau
  6. P. Gérard, The Cauchy problem for the Gross–Pitaevskii equation
  7. Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Motion of concentration sets in Ginzburg-Landau equations
  8. Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Improved estimates for the Ginzburg-Landau equation : the elliptic case

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