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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.

MSC:
 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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References:
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