Asymptotic Stability of Precessing Domain Walls for the Landau–Lifshitz–Gilbert Equation in a Nanowire with Dzyaloshinskii–Moriya Interaction

• Published in: Communications in mathematical physics Vol. 401; no. 3; pp. 2901 - 2957
• Main Authors: Côte, Raphaël, Ignat, Radu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023; Springer Nature B.V; Springer Verlag
• Subjects: Analysis of PDEs; Asymptotic properties; Classical and Quantum Gravitation; Complex Systems; Critical point; Domain walls; Ferromagnetism; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Nanowires; Perturbation; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Stability; Theoretical; Transition layers; Translations
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-023-04714-9

We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via Γ -convergence that determines a limit functional E defined for maps m : R → S 2 in the direction e 1 of the nanowire. The energy functional E is invariant under translations in e 1 and rotations about the axis e 1 . We fully classify the critical points of finite energy E when a transition between - e 1 and e 1 is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau–Lifshitz–Gilbert equation associated to E under the effect of an applied magnetic field h ( t ) e 1 depending on the time variable t gives rise to the so-called precessing domain wall . Our main result proves the asymptotic stability of precessing domain walls for small h in L ∞ ( [ 0 , + ∞ ) ) and small H 1 ( R ) perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional E .