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Nonlinear Stability of Static Néel Walls in Ferromagnetic Thin Films

A. Capella, C. Melcher, L. Morales, R. G. Plaza

Abstract

In this paper, the nonlinear (orbital) stability of static 180^\circ Néel walls in ferromagnetic films, under the reduced wave-type dynamics for the in-plane magnetization proposed by Capella, Melcher and Otto [CMO07], is established. It is proved that the spectrum of the linearized operator around the static Néel wall lies in the stable complex half plane with non-positive real part. This information is used to show that small perturbations of the static Néel wall converge to a translated orbit belonging to the manifold generated by the static wall.

Nonlinear Stability of Static Néel Walls in Ferromagnetic Thin Films

Abstract

In this paper, the nonlinear (orbital) stability of static 180^\circ Néel walls in ferromagnetic films, under the reduced wave-type dynamics for the in-plane magnetization proposed by Capella, Melcher and Otto [CMO07], is established. It is proved that the spectrum of the linearized operator around the static Néel wall lies in the stable complex half plane with non-positive real part. This information is used to show that small perturbations of the static Néel wall converge to a translated orbit belonging to the manifold generated by the static wall.
Paper Structure (24 sections, 39 theorems, 222 equations)

This paper contains 24 sections, 39 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. The micromagnetic model
  4. Stationary Néel wall profile in soft magnetic thin films
  5. LLG dynamics
  6. LLG wave-type dynamic thin film limit
  7. Main result
  8. Strategy of the proof
  9. The linearized operator around the static Néel wall's phase
  10. Basic properties
  11. The spectrum of ${\mathcal{L}}$
  12. The asymptotic operator ${\mathcal{L}}_\infty$
  13. Relative compactness
  14. Perturbation equations and spectral stability
  15. The perturbation equation
  16. ...and 9 more sections

Key Result

Proposition 2.1

There exists a static Néel wall solution with phase $\overline{\theta} = \overline{\theta}(x)$, $\overline{\theta} : \mathbb{R} \to (-\pi/2,\pi/2)$, satisfying the following:

Theorems & Definitions (79)

  • Proposition 2.1: properties of the static Néel wall's phase CMO07Melc03
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3: Orbital stability of the static Néel wall
  • Remark 2.4
  • Remark 2.5
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • ...and 69 more