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Rigorous results on Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field. II. (English) Zbl 0869.34016
The study of superconductivity for models in a film is reduced to the problem \[ \begin{aligned} -\kappa^{-2}f''- f+ f^3+ A^2f &= 0,\qquad f'(\pm d/2)=0,\tag{1}\\ - A''+ f^2A & = 0,\qquad A'(\pm d/2)=h.\end{aligned} \] In a previous paper [Nonlinear Stud. 3, No. 1, 1-29 (1996; Zbl 0857.34006)], the authors establish basic properties of (1) for \(\kappa=0\), \(d=0\). In this paper, the asymptotic behavior of different critical fields is determined (\(\kappa\) small, \(d\) small) by means of the results in Part I. Moreover, general properties of the Ginzburg-Landau equation (1) are derived. The cases \(h=0\) (no external magnetic field) and \(h\) small are investigated in detail.

34B15 Nonlinear boundary value problems for ordinary differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
78A35 Motion of charged particles
34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems