# The cubic Schr{\"o}dinger regime of the Landau-Lifshitz equation with a strong easy-axis anisotropy

## TL;DR

This paper rigorously establishes the connection between the Landau-Lifshitz equation and the cubic Schrödinger equation in the strong easy-axis anisotropy regime, including soliton classification and convergence analysis.

## Contribution

It provides a rigorous proof of the cubic Schrödinger regime for the Landau-Lifshitz equation and classifies one-dimensional solitons with convergence results.

## Key findings

- Proved the cubic Schrödinger approximation in any dimension.
- Classified one-dimensional solitons of the Landau-Lifshitz equation.
- Quantified convergence of solitons to cubic Schrödinger solitons.

## Abstract

We pursue our work on the asymptotic regimes of the Landau-Lifshitz equation for bi-axial ferromagnets. We put the focus on the cubic Schr{\"o}dinger equation, which is known to describe the dynamics 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 Schr{\"o}dinger equation.