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On a family of solutions of the second Painlevé equation related to superconductivity

Published online by Cambridge University Press:  01 June 1998

Bernard Helffer  and
Fred B. Weissler
Bernard Helffer
Affiliation:
Département de Mathématiques, Université Paris-Sud, UA 760 du CNRS, Bat. 425, F-91405 Orsay Cedex, France
Fred B. Weissler
Affiliation:
Laboratoire Analyse, Géométrie et Applications, Université Paris-Nord, UMR 7539 du CNRS, Institut Galilée, F-93430 Villetaneuse, France
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Abstract

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In a recent paper devoted to the study of the superheating field attached to a semi-infinite superconductor, Chapman [1] constructs a family of approximate solutions of the Ginzburg–Landau system. This construction, based on a matching procedure, implicitly uses the existence of a family of solutions depending on a parameter c∈IR of the Painlevé equation in a semi-infinite interval (0, +∞)

with a Neumann condition at 0

and having a prescribed behaviour at +∞

In this paper we prove the existence of such a family of solutions and investigate its properties. Moreover, we prove that the second coefficient in Chapman's expansion of the superheating field is finite.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 9 , Issue 3 , June 1998 , pp. 223 - 243
Copyright
Copyright © 1998 Cambridge University Press