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

Harini Desiraju, Alexander R. Its, Andrei Prokhorov

TL;DR

This work develops a genus-one extension of the nonlinear steepest descent method to analyze the Landau-Lifshitz equation on the torus, solving the Cauchy problem in the soliton-free Schwartz class. It starts from the direct-scattering map and formulates a genus-one Riemann-Hilbert problem on the torus, then constructs a global and a local parametrix to extract a rigorous large-time asymptotics. The main result provides leading-order decay with explicit phase $\theta(x,t)$ and amplitude for the LL components $L_1,L_2,L_3$ in the regime $0<m\le\kappa=x/t\le M$, including a parabolic-cylinder-based local model at the stationary point $\lambda_0$. The approach extends previous genus-zero techniques and offers a framework for rigorous asymptotics of other integrable PDEs on genus-one surfaces, enabling systematic analysis on toroidal backgrounds.

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.

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

TL;DR

This work develops a genus-one extension of the nonlinear steepest descent method to analyze the Landau-Lifshitz equation on the torus, solving the Cauchy problem in the soliton-free Schwartz class. It starts from the direct-scattering map and formulates a genus-one Riemann-Hilbert problem on the torus, then constructs a global and a local parametrix to extract a rigorous large-time asymptotics. The main result provides leading-order decay with explicit phase and amplitude for the LL components in the regime , including a parabolic-cylinder-based local model at the stationary point . The approach extends previous genus-zero techniques and offers a framework for rigorous asymptotics of other integrable PDEs on genus-one surfaces, enabling systematic analysis on toroidal backgrounds.

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.
Paper Structure (15 sections, 17 theorems, 236 equations, 8 figures)

This paper contains 15 sections, 17 theorems, 236 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Direct scattering problem
  3. Inverse scattering problem
  4. Singular integral equation for Riemann-Hilbert problem on the torus
  5. Reconstruction of the linear equation \ref{['def:LaxU']} from $Y(\lambda, x)$
  6. Turning on time
  7. Nonlinear steepest descent analysis
  8. Global parametrix
  9. Local parametrix
  10. Small norm problem
  11. Proof of Theorem \ref{['thm:asymp_LL']}
  12. Jacobi elliptic functions
  13. Weierstrass zeta function
  14. Parabolic cylinder parametrix
  15. Proofs of Propositions \ref{['Prop:Rodin']}, \ref{['prop:int_ab']}, and \ref{['prop:a_b_smooth']}

Key Result

Theorem 1.1

Let $Y(\lambda, x)$ be the solution of the Riemann-Hilbert problem rhp1 with ${\sf r}(\lambda)$ satisfying properties r_property_1-r_property_6 of section sec:direct. Then the function $L(x)$ constructed from it by formula (see also sol_LL_RHP_init) belongs to the Schwartz class: $L_1(x), L_2(x), L_3(x)-1 \in \mathcal{S}(\mathbb{R})$, and it defines the initial data for the LL equation whose refl

Figures (8)

  • Figure 1: Torus.
  • Figure 3: Take $\varkappa=1$, $k=\frac{1}{2}$. We draw the following sign chart: the red and green parts denote the negative and positive domains of the imaginary part of the characteristic exponent $\mathrm{Im} (p(\lambda,\varkappa))$. We also draw the stokes lines $\mathrm{Re} (p(\lambda,\varkappa)=p(\lambda_0,\varkappa)$ centered at $\lambda_0$, $\lambda_0+2K$ with blue color and the stokes lines $\mathrm{Re} (p(\lambda,\varkappa)=-p(\lambda_0,\varkappa)$ centered at $\lambda_0+2\mathrm{i} K'$, $\lambda_0+2K+2\mathrm{i} K'$ with orange color.
  • Figure 4: Jump contour $\Sigma$ on the fundamental domain.
  • Figure 5: Contour $\Sigma(\lambda_0)$ in the sub-domain $\mathbb{T}^2(\lambda_0)$.
  • Figure 6: Contour $\Sigma^{(loc)}(\lambda_0)$ and the respective jumps in the disc $\mathbb{D}(\lambda_0)$.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • proposition 1: rodin1984
  • proof
  • proposition 2: rodin1984
  • proposition 3
  • proposition 4
  • proposition 5
  • proof
  • Remark 2.1
  • ...and 24 more