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Scattering for the dispersion managed nonlinear Schrödinger equation

Mi-Ran Choi, Kiyeon Lee, Young-Ran Lee

Abstract

We consider the dispersion managed nonlinear Schrdinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing the $U^p$ and $V^p$ spaces.

Scattering for the dispersion managed nonlinear Schrödinger equation

Abstract

We consider the dispersion managed nonlinear Schrdinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in for small initial data employing the and spaces.
Paper Structure (6 sections, 10 theorems, 84 equations)

This paper contains 6 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. $U^p-V^p$ and adapted function spaces
  5. Bilinear estimates
  6. Proof of Theorem \ref{['thm:scattering']}

Key Result

Theorem 1.1

Assume ${d_{\mathrm{av}}}\neq0$ and $\sigma =2/d$ for $d=1,2$. Given initial data $u_0\in L_x^2(\mathbb{R}^d)$ with small enough $L_x^2$-norm, the Cauchy problem eq:main is globally well-posed in $L_x^2(\mathbb{R}^d)$. Moreover, the solution $u(t)$ scatters in $L_x^2(\mathbb{R}^d)$, that is, there e

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2: Corollary 4.24. of kotavi-book
  • Lemma 2.3
  • proof
  • Lemma 2.4: Logarithmic interpolation
  • Lemma 2.5: Transference principle
  • Lemma 3.1: Strichartz estimate
  • Lemma 3.2: Bilinear Strichartz estimate
  • ...and 8 more