Oscillatory buckling mode in thin-film nucleation

We are interested in switching of elongated thin-film elements as described by the micromagnetic model. Nucleation occurs when the saturation branch becomes unstable at a critical field. It is characterized by a degeneracy of the Hessian of the micromagnetic energy. The degenerate subspace describes the unstable mode. In a prior work [Cantero-Alvarez and Otto, Journal of Nonlinear Science], we showed that there are four regimes for nucleation. We proved this by identifying the scaling of the critical field in the non-dimensional parameters. This contradicts a claim by Aharoni that there are at most three regimes. Two of these regimes are buckling regimes, where the magnetization is pinned at the lateral facets of the sample and buckles in-plane. As shown in our prior work, these regimes differ in the scaling of the critical field. Here we show that the unstable modes are also qualitatively different: Only in one of the regimes do they oscillate in the long direction. The unstable modes are asymptotically identified by iota-convergence, which is applied to the Rayleigh quotient of the Hessian.