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Rigorous derivation of an effective model for periodic Schrödinger equations with linear band crossing of Dirac type

Elena Danesi

Abstract

In this paper we consider a family of time-dependent 1-dimensional cubic Schrödinger equation (NLS) with periodic potential. Exploiting semiclassical scaling and multiscale analysis, we derive an effective nonlinear Dirac equation, which describes the dynamics of solutions to NLS spectrally localized around Dirac points.

Rigorous derivation of an effective model for periodic Schrödinger equations with linear band crossing of Dirac type

Abstract

In this paper we consider a family of time-dependent 1-dimensional cubic Schrödinger equation (NLS) with periodic potential. Exploiting semiclassical scaling and multiscale analysis, we derive an effective nonlinear Dirac equation, which describes the dynamics of solutions to NLS spectrally localized around Dirac points.
Paper Structure (11 sections, 7 theorems, 79 equations)

This paper contains 11 sections, 7 theorems, 79 equations.

Table of Contents

  1. Introduction and main result
  2. Strategy
  3. Organization of the paper
  4. Preliminaries
  5. Notation
  6. Floquet--Bloch theory
  7. Dirac points
  8. Functional background
  9. Two--scale asymptotic expansion
  10. Local well--posedness of the NLD
  11. Proof of the result

Key Result

Theorem 1.1

Let $V$ satisfies Assumption ass:V and let $(\pi, \mu_*)$ be a Dirac point for the operator $H \coloneqq -\partial_x^2 + V(x)$. Let $\Phi_-(\cdot,k), \Phi_+(\cdot,k)$, $k \in [0,2\pi]$ be the two corresponding Bloch waves as in Section subsec:Dirac_points. Let Moreover, let $\alpha \in C([0,T]; H^S(\mathbb{R}))^2$ be a solution to the nonlinear Dirac equation NLD for some $S > s+3$, $s>\frac{1}{2

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7: Gagliardo--Nirenberg inequality
  • proof
  • Lemma 2.8: Moser--type Lemma
  • Remark 3.1
  • ...and 10 more