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Renormalized energy of interacting Ginzburg–Landau vortex filaments
Author(s) -
Del Pino Manuel,
Kowalczyk Michał
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm126
Subject(s) - vortex , physics , cylinder , degenerate energy levels , logarithm , action (physics) , classical mechanics , mathematical physics , mathematical analysis , quantum mechanics , mathematics , geometry , mechanics
We consider the Ginzbug–Landau energy in a cylinder in ℝ 3 , and a canonical approximation for critical points with an assembly of n ⩾2 periodic vortex lines near the axis of the cylinder. We find a formula for the energy which, up to a large additive constant and to leading order, is the action functional of the n ‐body problem with a logarithmic potential in ℝ 2 , the axis variable playing the role of time. A special family of rotating helicoidal critical points of the functional is found to be non‐degenerate up to the invariances of the problem, and therefore persistent under small perturbations. Our analysis suggests the presence of very complex stationary configurations for vortex filaments, potentially also involving intersecting filaments.