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Large scale limit for a dispersion-managed NLS

Jason Murphy

TL;DR

This work establishes a rigorous large-scale limit connecting the Gabitov--Turitsyn dispersion-managed NLS to the standard power-type NLS in the defocusing, two-dimensional cubic regime. By scaling the NLS solution and exploiting a bespoke stability theory for DMNLS built on shifted Strichartz estimates, the authors show that DMNLS solutions with suitably large spatial scales approximate NLS solutions, and vice versa in the small-dispersion limit. The main result proves that for any initial data in $L^2$, there exist global, scattering DMNLS solutions at sufficiently small scaling parameter, and that these DMNLS solutions converge to the corresponding scaled NLS solutions in both the $L_t^{\infty}L_x^2$ and spacetime $L_{t,x}^4$ norms. Consequently, the DMNLS model admits global, scattering solutions of arbitrarily large $L^2$-norm, linking averaging phenomena in dispersion management to classical NLS dynamics and enabling insights into both large- and small-scale limits.

Abstract

We derive the standard power-type NLS as a scaling limit of the Gabitov--Turitsyn dispersion-managed NLS, using the $2d$ defocusing, cubic equation as a model case. In particular, we obtain global-in-time scattering solutions to the dispersion-managed NLS for large scale data of arbitrary $L^2$-norm.

Large scale limit for a dispersion-managed NLS

TL;DR

This work establishes a rigorous large-scale limit connecting the Gabitov--Turitsyn dispersion-managed NLS to the standard power-type NLS in the defocusing, two-dimensional cubic regime. By scaling the NLS solution and exploiting a bespoke stability theory for DMNLS built on shifted Strichartz estimates, the authors show that DMNLS solutions with suitably large spatial scales approximate NLS solutions, and vice versa in the small-dispersion limit. The main result proves that for any initial data in , there exist global, scattering DMNLS solutions at sufficiently small scaling parameter, and that these DMNLS solutions converge to the corresponding scaled NLS solutions in both the and spacetime norms. Consequently, the DMNLS model admits global, scattering solutions of arbitrarily large -norm, linking averaging phenomena in dispersion management to classical NLS dynamics and enabling insights into both large- and small-scale limits.

Abstract

We derive the standard power-type NLS as a scaling limit of the Gabitov--Turitsyn dispersion-managed NLS, using the defocusing, cubic equation as a model case. In particular, we obtain global-in-time scattering solutions to the dispersion-managed NLS for large scale data of arbitrary -norm.

Paper Structure

This paper contains 4 sections, 5 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability Theory for DMNLS
  4. Convergence

Key Result

Theorem 1.1

Let $\varphi \in L^2(\mathbb{R}^2)$, and let $u$ be the global, scattering solution to nls with $u|_{t=0}=\varphi$. For $\lambda>0$, let For all $\lambda$ sufficiently small, the solution $v_\lambda$ to dmnls with initial data $\varphi_\lambda$ exists globally in time and scattersWe say a solution $v$scatters in $L^2$ if there exists $v_\pm\in L^2$ such that \lim_{t\to\pm\infty}\|v(t)-e^{it\Delta

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 2.1: Shifted Strichartz estimate, KawakamiMurphy
  • proof
  • Theorem 3.1: Local existence
  • proof
  • Lemma 3.1: Short-time stability
  • proof
  • Theorem 3.2: Long-time stability
  • proof
  • Remark 3.1
  • ...and 1 more