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Expansion for the superheating field in a semi-infinite film in the weak-κ limit

Published online by Cambridge University Press:  15 January 2003

Pierre Del Castillo
Pierre Del Castillo*
Affiliation:
UMR 8628 du CNRS, Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France. pierre.castillo@math.u.psud.fr.
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Abstract

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-κ limit. These authors deduceda formal expansion for the superheating field in powers of $\kappa^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers of $\kappa^{\frac{1}{2}}$ for the superheating field.

Keywords

SuperconductivityGinzburg-Landau equationcritical field.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 971 - 993
Copyright
© EDP Sciences, SMAI, 2002

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References

C. Bolley and P. Del Castillo, Existence and uniqueness for the half-space Ginzburg-Landau model. Nonlinear Anal. 47/1 (2001) 135-146.
Bolley, C. and Helffer, B., Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak κ-limit. Rev. Math. Phys. 8 (1996) 43-83. CrossRef
Bolley, C. and Helffer, B., Rigorous results on the Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. Part II. Nonlinear Stud. 3 (1996) 1-32.
Bolley, C. and Helffer, B., Proof of the De Gennes formula for the superheating field in the weak-κ limit. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 597-613. CrossRef
Bolley, C. and Helffer, B., Superheating in a semi-infinite film in the weak-κ limit: Numerical results and approximate models. ESAIM: M2AN 31 (1997) 121-165. CrossRef
Bolley, C. and Helffer, B., Upper bound for the solution of the Ginzburg-Landau equations in a semi-infinite superconducting field and applications to the superheating field in the large κ regime. European J. Appl. Math. 8 (1997) 347-367.
Bolley, C., Foucher, F. and Helffer, B., Superheating field for the Ginzburg-Landau equations in the case of a large bounded interval. J. Math. Phys. 41 (2000) 7263-7289. CrossRef
Chapman, S., Superheating field of type II superconductors. SIAM J. Appl. Math. 55 (1995) 1233-1258. CrossRef
P. Del Castillo, Thèse de doctorat. Université Paris-Sud (2000).
P. Del Castillo, Two terms in the lower bound for the superheating field in a semi-infinite film in the weak-κ limit. European J. Appl. Math. (2002).
P.G. De Gennes, Superconductivity: Selected topics in solid state physics and theoretical Physics, in Proc. of 8th Latin american school of physics. Caracas (1966).
Ginzburg, V.L., On the theory of superconductivity. Nuovo Cimento 2 (1955) 1234. CrossRef
Ginzburg, V.L., On the destruction and the onset of superconductivity in a magnetic field. Zh. Èksper. Teoret. Fiz. 34 (1958) 113-125; Transl. Soviet Phys. JETP 7 (1958) 78-87.
Di Bartolo, T. Dorsey and J. Dolgert, Superheating fields of superconductors: Asymptotic analysis and numerical results. Phys. Rev. B 53 (1996); Erratum. Phys. Rev. B 56 (1997).
W. Eckhaus, Matched asymptotic expansions and singular perturbations. North-Holland, Math. Studies 6 (1973).
Helffer, B. and Weissler, F., On a family of solutions of the second Painlevé equation related to superconductivity. European J. Appl. Math. 9 (1998) 223-243. CrossRef
Parr, H., Superconductive superheating field for finite κ. Z. Phys. B 25 (1976) 359-361. CrossRef
M. van Dyke, Perturbation Methods in fluid mechanics. Academic Press, Stanford CA (1975).
B. Rothberg-Bibby, H.J. Fink and D.S. McLachlan, First and second order phase transitions of moderately small superconductor in a magnetic field. North-Holland (1978).
S. Kaplun, Fluid mechanics and singular perturbations. Academic Press (1967).