The cubic Schrödinger regime of the Landau–Lifshitz equation with a strong easy-axis anisotropy

We pursue our work on the asymptotic regimes of the Landau–Lifshitz equation. We put the focus on the cubic Schrödinger equation, which is known to describe the dynamics of uniaxial ferromagnets in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of the Landau–Lifshitz equation and quantify their convergence towards the solitons of the one-dimensional cubic Schrödinger equation. Our proof is based on a nonlinear rewriting of the Landau–Lifshitz equation for uniaxial ferromagnets as a nonlinear Schrödinger equation. This latter equation turns out to be consistent with the cubic Schrödinger equation in a regime of strong easy-axis anisotropy. We rely on this consistency to perform suitable energy estimates and then control the difference between the solutions of the two equations.