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On dispersion managed nonlinear Schrödinger equations with lumped amplification

Mi-Ran Choi, Younghoon Kang, Young-Ran Lee

Abstract

We show the global well-posedness of the nonlinear Schrödinger equation with periodically varying coefficients and a small parameter $\varepsilon>0$, which is used in optical-fiber communications. We also prove that the solutions converge to the solution for the Gabitov-Turitsyn or averaged equation as $\varepsilon$ tends to zero.

On dispersion managed nonlinear Schrödinger equations with lumped amplification

Abstract

We show the global well-posedness of the nonlinear Schrödinger equation with periodically varying coefficients and a small parameter , which is used in optical-fiber communications. We also prove that the solutions converge to the solution for the Gabitov-Turitsyn or averaged equation as tends to zero.

Paper Structure

This paper contains 4 sections, 10 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Well--posedness
  3. Averaging theorem
  4. Linear propagator

Key Result

Theorem \oldthetheorem

Let ${d_{\mathrm{av}}}\in {\mathbb{R}}$. For every $u_0 \in H^1({\mathbb{R}})$, there exists a unique solution $u\in \mathcal{C}({\mathbb{R}}, H^1({\mathbb{R}}))$ of eq:intro_main. Moreover, $u$ depends continuously on the initial datum in the following sense. For every $M>0$, the map $u_0 \mapsto u

Theorems & Definitions (21)

  • Theorem \oldthetheorem: Global well--posedness
  • Theorem \oldthetheorem: Averaging Theorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • proof : Proof of Proposition \ref{['prop:H^slocal theory']}
  • proof : Proof of Corollary \ref{['cor:maximal existence in H^1']}
  • Proposition \oldthetheorem
  • proof
  • ...and 11 more