Modélisation du champ de retard a la condensation d’un supraconducteur par un problème de bifurcation. (Modelling of the condensation delay field of a supraconductor by a bifurcation problem). (French) Zbl 0741.35085

Bolley, C.

RAIRO, Modélisation Math. Anal. Numér. 26, No. 2, 235-287 (1992).

Summary: A study of Ginzburg-Landau equations, describing the different states of a superconducting film of thickness a, has shown the existence of solutions bifurcating from all the trivial solutions which satisfy two necessary bifurcating conditions. Let \(({\mathcal P}_ c)\) be the problem defined by these two conditions. Problem \(({\mathcal P}_ c)\) is studied as a branching problem: first, a study of trivial solutions in paragraph 2.2, followed by a study of a nontrivial solutions in paragraph 2.3. For this last study, we write three equivalent necessary bifurcating conditions for \(({\mathcal P}_ c)\); and we show that one of them is satisfied for, at least, one value of the parameter \(a\). An hypothesis of transversality is added to guarantee the existence of bifurcating solutions. This hypothesis is verified in numerical results. However, a direct proof, without this hypothesis, shows that nontrivial solutions exist when \(a\) is large enough.
Using stability arguments, section III shows, which solutions of \(({\mathcal P}_ c)\) give the super-cooling field of the superconducting film.

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
34C23	Bifurcation theory for ordinary differential equations
35B32	Bifurcations in context of PDEs

Keywords:

Ginzburg-Landau equations; bifurcating conditions; existence of bifurcating solutions

References:

[1]	J. BLOT, Relation entre les grandeurs supraconductrices caractéristiques de l’aluminium massif et les champs de transition de films divisés, en fonction de leur épaisseur. Thèse soutenue à Rennes 1, 1987.
[2]	C. BOLLEY, Bifurcations dans les équations de Ginzburg-Landau des matériaux supraconducteurs soumis à un champ magnétique extérieur. Publications de l’E.N.S.M. 1988.
[3]	[3] C. BOLLEY, Familles de branches de bifurcations dans les équations de Ginzburg-Landau, RAIRO M2AN Vol. 25, n^\circ 3, 1991. Zbl0726.34031 MR1103091 · Zbl 0726.34031
[4]	M. G. CRANDALL and P. H. RABINOWITZ, Bifurcation from Simple Eigenvalues, J. Funct. Anal. 8, 1971. Zbl0219.46015 MR288640 · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[5]	M. DAUGE and B. HELFFER, Eigenvalues variations I. Neumann problem for Sturm-Liouville operators, à paraître. Zbl0784.34021 · Zbl 0784.34021 · doi:10.1006/jdeq.1993.1071
[6]	B. DUGNOILLE, Étude théorique et expérimentale des propriétés magnétiques des couches minces supraconductrices de type I et de kappa faibles. Thèse soutenue à Mons, 1978.
[7]	V. L. GINZBURG, Soviet Physics JETP 7, 78, 1958. MR101069
[8]	B. HELFFER, Communication personnelle.
[9]	B. HELFFER, Semi-classical Analysis for the Schrödinger Operator and Applications. Lecture Notes in Math. n^\circ 1336, 1980. Zbl0647.35002 · Zbl 0647.35002
[10]	T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, n^\circ 132, 1976. Zbl0342.47009 MR407617 · Zbl 0342.47009
[11]	M. KRASNOSEL’SKII, Topological Methods in the Theory of Nonlinear Intégral Eq., Pergamon Press, 1964. Zbl0111.30303 · Zbl 0111.30303
[12]	B. M. LEVITAN and I. S. SARGSJAN, Introduction to Spectral Theory : Selfadjoint Ordinary Diff. Equations, American Math. Soc., 1975. Zbl0302.47036 MR369797 · Zbl 0302.47036
[13]	Y. PELLANÉtude de la métastabilité de la transition supraconductrice de films divisés d’indium sous champ magnétique parallèle et perpendiculaire. Thèse soutenue à Rennes 1, 1987.
[14]	P. H. RABINOWITZ, Some Global Results for Nonlinear Eigenvalue Problems. J. Funct. Anal., n^\circ 7, pp. 487-513, 1971. Zbl0212.16504 MR301587 · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[15]	D. ST JAMES et P. G. de GENNES, Phys. Lett. 7, 306, 1963.
[16]	L. SCHWARTZ, Méthodes mathématiques pour les sciences physiques, Hermann, Paris, 1965. Zbl0904.35001 MR143360 · Zbl 0904.35001
[17]	Y. SIBUYA, Global Theory of a Second Order Linear Differential Equation with a Polynomial Coefficient, North Holland, 1975. Zbl0322.34006 · Zbl 0322.34006
[18]	J. SMOLLER, Schock Waves and Reaction Diffusion Equations, n^\circ 258. Springer-Verlag, 1980. Zbl0807.35002 MR1301779 · Zbl 0807.35002