2013/09/10 by Döring, Lukas, Ignat, Radu, Otto, Felix
#35B32 #35B36 #49J45 #49S05 #78A30 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1309.2494
We study the Landau-Lifshitz model for the energy of multi-scale transition layers -- called "domain walls" -- in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors m^± ∈ \mathbbS2 that differ by an angle 2α. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in \mathbbS2 by an angle 2θ, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle θ to α in \mathbbS1. The angle θ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7]. Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).