Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof

被引：8

作者：

Correc, Anais ^{[1
]}

Lessard, Jean-Philippe ^{[1
]}

机构：

[1] Univ Laval, Dept Math & Stat, Quebec City, PQ G1V 0A6, Canada

来源：

EUROPEAN JOURNAL OF APPLIED MATHEMATICS | 2015年 / 26卷

关键词：

Ginzburg-Landau equation; boundary value problems; coexistence of nontrivial solutions; rigorous numerics; Chebyshev series; contraction mapping theorem; RIGOROUS NUMERICS; VALIDATED CONTINUATION; BIFURCATION DIAGRAM; ASYMPTOTIC ANALYSIS; EQUILIBRIA; SYSTEM; MODEL;

D O I：

10.1017/S0956792514000308

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.

引用

页码：33 / 60

页数：28

共 33 条

[1] On the minimizers of the Ginzburg-Landau energy for high kappa: the one-dimensional case [J].

Aftalion, A .

EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 1997, 8 :331-345

[2] Asymptotic analysis of the bifurcation diagram for symmetric one-dimensional solutions of the Ginzburg-Landau equations [J].

Aftalion, A ;

Chapman, SJ .

EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 1999, 10 :477-495

[3] On the uniqueness of solutions of the Ginzburg-Landau system for thin films [J].

Aftalion, A ;

Troy, WC .

EUROPEAN JOURNAL OF APPLIED MATHEMATICS, 2000, 11 :365-380

[4] On the solutions of the one-dimensional Ginzburg-Landau equations for superconductivity [J].

Aftalion, A ;

Troy, WC .

PHYSICA D, 1999, 132 (1-2) :214-232

[5] The bifurcation diagrams for the Ginzburg-Landau system of superconductivity [J].

Aftalion, A ;

Du, Q .

PHYSICA D-NONLINEAR PHENOMENA, 2002, 163 (1-2) :94-105

[6] Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity [J].

Aftalion, A ;

Chapman, SJ .

SIAM JOURNAL ON APPLIED MATHEMATICS, 2000, 60 (04) :1157-1176

[7]

[Anonymous], 1999, Developments in reliable computing, DOI DOI 10.1007/978-94-017-1247-7

[8] Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto-Sivashinski Equation [J].

Arioli, Gianni ;

Koch, Hans .

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2010, 197 (03) :1033-1051

[9]

Boyd J., 2001, Chebyshev and fourier spectral methods, V2nd

[10] Global Bifurcation Diagrams of Steady States of Systems of PDEs via Rigorous Numerics: a 3-Component Reaction-Diffusion System [J].

Breden, Maxime ;

Lessard, Jean-Philippe ;

Vanicat, Matthieu .

ACTA APPLICANDAE MATHEMATICAE, 2013, 128 (01) :113-152

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