Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation
For each given n ≥ 2, we construct a family of entire solutions u"(z, t), ε > 0, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation ∆u ＋ (1 - ⃒u⃒2)u ＝ 0, (z, t) ∊ R2 × R ≃ R3. These solutions are 2π=ε-periodic in t and have n helix-vortex curves, with asym...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/709208 |
| Acceso en línea: | http://hdl.handle.net/10486/709208 |
| Access Level: | acceso abierto |
| Palabra clave: | Ginzburg–landau equation Logarithmic n-body problem Renormalized energy Vortex filamnets Matemáticas |
| Sumario: | For each given n ≥ 2, we construct a family of entire solutions u"(z, t), ε > 0, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation ∆u ＋ (1 - ⃒u⃒2)u ＝ 0, (z, t) ∊ R2 × R ≃ R3. These solutions are 2π=ε-periodic in t and have n helix-vortex curves, with asymptotic behavior, as ε → 0, (Formular Presented) where W (z) ＝ w(r)ei θ , z ＝ r ei θ , is the standard degree ＋1 vortex solution of the planar Ginzburg–Landau equation ∆W ＋ (1 - ⃒W ⃒2)W ＝ 0 in R2 and (Formular Presented) Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), f (t) ＝(f1(t), . . ., fn(t)) being a rotating equilibrium point for the renormalized energy of vortex filaments derived there,(Formular Presented) corresponding to that of a planar logarithmic n-body problem. The modulus of these solutions converges to 1 as ⃒z⃒ goes to infinity uniformly in t, and the solutions have nontrivial dependence on t, thus negatively answering the Ginzburg–Landau analogue of the Gibbons conjecture for the Allen–Cahn equation, a question originally formulated by H. Brezis. © 2022 European Mathematical Society Published by EMS Press |
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