Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation

TL;DR

This paper extends the nonlinear steepest descent method to genus 1 surfaces to analyze the long-term behavior of the Landau-Lifshitz equation on a torus, providing a framework for studying other integrable equations in similar settings.

Contribution

It introduces a novel application of nonlinear steepest descent on a torus for the Landau-Lifshitz equation, enabling rigorous large-time asymptotic analysis.

Findings

Established large-time asymptotics for the Landau-Lifshitz equation on a torus.

Extended the nonlinear steepest descent method to genus 1 surfaces.

Provided a foundation for analyzing other integrable equations on the torus.

Abstract

We obtain rigorous large time asymptotics for the Landau-Lifshitz equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the Landau-Lifshitz equation.