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

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.


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