The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

  • André de Laire

    Université de Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
  • Philippe Gravejat

    Université de Cergy-Pontoise, Laboratoire de Mathématiques (UMR 8088), F-95302 Cergy-Pontoise Cedex, France
The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy cover
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Abstract

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.

This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.

Cite this article

André de Laire, Philippe Gravejat, The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, pp. 1885–1945

DOI 10.1016/J.ANIHPC.2018.03.005