The Concertina Pattern: A Bifurcation in Ferromagnetic Thin Films
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The concertina pattern is a metastable stage in the switching process of elongated thin-film elements. It is an approximately periodic structure of domains, separated by walls perpendicular to the long axis of the element. In this paper, we give arguments in favor of our claim that the period is frozen-in at nucleation, i.e., at the critical external field. In prior work, R. Cantero-Alvarez and F. Otto, Journal of Nonlinear Science (2006), we argued that there are four qualitatively different regimes for nucleation. In one of these asymptotic regimes, the unstable mode displays an oscillatory behavior in the direction of the long axis. In this work, we derive a scaling limit of the micromagnetic energy near the bifurcation point in the above regime. We also prove that the scaling limit is coercive for all values of the reduced external field. Because of this coercivity, there exists a branch of nontrivial local minimizers. Numerical minimization of the scaling limit reveals that this branch is indeed a continuous branch of concertina pattern. The scaling limit is derived by Γ-convergence of the suitably rescaled micromagnetic energy. This robust procedure combines the limit of an asymptotic parameter regime with a zoom-in in configuration space. The coercivity of the scaling limit is derived by suitable nonlinear interpolation estimates.
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