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Lattice Approximations to NLS

Zhimeng Ouyang

Abstract

In this paper, we prove that solutions of the discrete NLS lattice model for $L^2$ initial data with double frequency components converge to solutions of a coupled system of cubic NLS.

Lattice Approximations to NLS

Abstract

In this paper, we prove that solutions of the discrete NLS lattice model for initial data with double frequency components converge to solutions of a coupled system of cubic NLS.

Paper Structure

This paper contains 19 sections, 15 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Initial Data Sampling
  4. Solution Reconstruction
  5. Main Result
  6. Method
  7. Preliminaries
  8. Notation and Convention
  9. Norms
  10. Passage Between Lattice and Continuum Functions
  11. Dispersive and Strichartz Estimates
  12. Conserved Quantities and Well-posedness
  13. Almost Conservation Law
  14. Persistence of Schrödinger-like Dynamics
  15. Precompactness
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $(\psi_0,\phi_0)\in L^2(\mathbb{R})$ and let $(\psi,\phi)\in (C_tL^2_x\cap L_{t,loc}^6L_x^6)(\mathbb{R}\times\mathbb{R})$ denote the unique global solution of the coupled system NLS_low-NLS_high with this initial data. Given $0<h\leq h_0\ll1$ sufficiently small, let $u_n(t) \in C_t\ell_n^2(\math moreover, this yields the long-wave limit of the discrete model: $(\psi,\phi)$ describes the small-

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.1
  • Definition 2.1
  • Lemma 2.1: Embedding for $\ell^p$ spaces
  • proof
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.2
  • ...and 23 more