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Asymptotic stability of 2-domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction

Raphaël Côte, Guillaume Ferriere

TL;DR

The paper proves the asymptotic stability of a two-domain-wall configuration for the Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya interaction in a nanowire, under small time-dependent forcing. It develops a space-localized modulation framework that splits the dynamics into two moving domain-wall gauges and a remainder, then proves coercivity and dissipation estimates for the localized energy around each wall. By coupling modulation, energy localization, and implicit-function gauge construction, the authors show that perturbations decay in $H^1$ and the solution converges to a sum of two decoupled domain walls plus a constant shift, with the gauges converging up to an invariant gauge. The method handles wall–wall interactions through precise exponential localization and provides a robust toolbox for multi-soliton stability analysis in chiral magnetic models, with mild assumptions on the external field.

Abstract

We consider a ferromagnetic nanowire, with an energy functional E with easy-axis in the direction $e_1$, and which takes into account the Dzyaloshinskii-Moriya interaction. We consider configurations of the magnetization which are perturbations of two well separated domain wall, and study their evolution under the Landau-Lifshitz-Gilbert flow associated to E. Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy E. Our analysis builds on the framework developed in [4], taking advantage that it is amenable to space localisation.

Asymptotic stability of 2-domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction

TL;DR

The paper proves the asymptotic stability of a two-domain-wall configuration for the Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya interaction in a nanowire, under small time-dependent forcing. It develops a space-localized modulation framework that splits the dynamics into two moving domain-wall gauges and a remainder, then proves coercivity and dissipation estimates for the localized energy around each wall. By coupling modulation, energy localization, and implicit-function gauge construction, the authors show that perturbations decay in and the solution converges to a sum of two decoupled domain walls plus a constant shift, with the gauges converging up to an invariant gauge. The method handles wall–wall interactions through precise exponential localization and provides a robust toolbox for multi-soliton stability analysis in chiral magnetic models, with mild assumptions on the external field.

Abstract

We consider a ferromagnetic nanowire, with an energy functional E with easy-axis in the direction , and which takes into account the Dzyaloshinskii-Moriya interaction. We consider configurations of the magnetization which are perturbations of two well separated domain wall, and study their evolution under the Landau-Lifshitz-Gilbert flow associated to E. Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy E. Our analysis builds on the framework developed in [4], taking advantage that it is amenable to space localisation.
Paper Structure (25 sections, 38 theorems, 255 equations)

This paper contains 25 sections, 38 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. A model for a ferromagnetic nanowire
  3. Functional spaces and Cauchy problem
  4. Statement of the main result
  5. On the assumptions on the external field h
  6. Proof of the stability
  7. Preliminary results
  8. Localisation and associated basis
  9. Interaction of domain walls
  10. Localisation
  11. Localized multilinear estimates in Sobolev spaces
  12. Coercivity of a Schrödinger operator
  13. Expansion in the associated basis
  14. Localised energies
  15. First estimate on the localised energies
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\alpha >0$, $\gamma \in (-1,1)$ and $h \in L^\infty([0,+\infty), \mathbb{R} )$. Assume $s \ge 1$ and $m_0 \in \mathcal{H}^s$. Then there exist a maximal time $T_+= T_+(m_0) \in (0, +\infty]$ and a unique solution $m \in\mathscr C([0, T_+), \mathcal{H}^s)$ to eq:llg with initial data $m_0$. More

Theorems & Definitions (64)

  • Theorem 1.1: Local well-posedness in $\mathcal{H}^s$
  • Theorem 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 2.1: Decomposition of the magnetization
  • Proposition 2.2: Coercivity of the energy
  • Proposition 2.3: Localised energy dissipation
  • proof : Proof of Theorem \ref{['th1']}, assuming Lemma \ref{['lem:decomp_magn']} and Propositions \ref{['prop:equiv_energy']} and \ref{['prop:est_dt_en']}
  • ...and 54 more