Ginzburg-Landau equations on non-compact Riemann surfaces

Nicholas M. Ercolani; Israel Michael Sigal; Jingxuan Zhang

doi:10.1016/j.jfa.2023.110074

Ginzburg-Landau equations on non-compact Riemann surfaces

Nicholas M. Ercolani, Israel Michael Sigal, Jingxuan Zhang^*

^*Corresponding author af dette arbejde

Institut for Matematiske Fag

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

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Abstract

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

Originalsprog	Engelsk
Artikelnummer	110074
Tidsskrift	Journal of Functional Analysis
Vol/bind	285
Udgave nummer	8
Antal sider	52
ISSN	0022-1236
DOI	https://doi.org/10.1016/j.jfa.2023.110074
Status	Udgivet - 2023

Bibliografisk note

Funding Information:
J. Zhang is supported by Danish National Research Foundation grant CPH-GEOTOP-DNRF151 , The Niels Bohr Grant of The Royal Danish Academy of Sciences and Letters , and NSERC Grant No. NA7901 . The research of IMS is supported in part by NSERC Grant No. NA7901 .

Publisher Copyright:
© 2023 The Author(s)

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10.1016/j.jfa.2023.110074Licens: CC BY

FulltextForlagets udgivne version, 824 KBLicens: CC BY

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Citationsformater

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abstract = "We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.",

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AU - Ercolani, Nicholas M.

AU - Sigal, Israel Michael

AU - Zhang, Jingxuan

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N2 - We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

AB - We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

KW - Bifurcation theory

KW - Elliptic equations on Riemann surfaces

KW - Ginzburg–Landau equations

KW - Superconductivity

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