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

André de Laire (1, 2) , Philippe Gravejat (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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.

Mots clés: Landau-Lifshitz equation; Long-wave approximation; Sine-Gordon equation

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

André de Laire, Philippe Gravejat. The Sine-Gordon regime of the Landau-Lifshitz equation with a strong easy-plane anisotropy. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, 35 (7), pp.1885-1945. ⟨10.1016/j.anihpc.2018.03.005⟩. ⟨hal-01518483⟩