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chebyshev series

European Journal of Applied Mathematics (1)

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Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof
ANAÏS CORREC, JEAN-PHILIPPE LESSARD
Journal:

European Journal of Applied Mathematics / Volume 26 / Issue 1 / February 2015

Published online by Cambridge University Press:

08 October 2014, pp. 33-60
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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.