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Continuum limit for discrete NLS with memory effect

Ricardo Grande

TL;DR

It is proved that solutions to this discrete Schrodinger equation converge strongly in L^2 to the solution to a continuous NLS-type equation with a memory effect, and the precise rate of convergence is computed.

Abstract

We consider a discrete nonlinear Schrödinger equation with long-range interactions and a memory effect on the infinite lattice $h\Z$ with mesh-size $h>0$. Such models are common in the study of charge and energy transport in biomolecules. Given that the distance between base pairs is small, we consider the continuum limit: a sharp approximation to the system as $h\rightarrow 0$. In this limit, we prove that solutions to this discrete equation converge strongly in $L^2$ to the solution to a continuous NLS-type equation with a memory effect, and we compute the precise rate of convergence. In order to obtain these results, we generalize some recent ideas proposed by Hong and Yang in $L^2$-based spaces to classical functional settings in dispersive PDEs involving the smoothing effect and maximal function estimates, as originally introduced in the pioneering works of Kenig, Ponce and Vega. We believe that our approach may therefore be adapted to tackle continuum limits of more general dispersive equations.

Continuum limit for discrete NLS with memory effect

TL;DR

It is proved that solutions to this discrete Schrodinger equation converge strongly in L^2 to the solution to a continuous NLS-type equation with a memory effect, and the precise rate of convergence is computed.

Abstract

We consider a discrete nonlinear Schrödinger equation with long-range interactions and a memory effect on the infinite lattice with mesh-size . Such models are common in the study of charge and energy transport in biomolecules. Given that the distance between base pairs is small, we consider the continuum limit: a sharp approximation to the system as . In this limit, we prove that solutions to this discrete equation converge strongly in to the solution to a continuous NLS-type equation with a memory effect, and we compute the precise rate of convergence. In order to obtain these results, we generalize some recent ideas proposed by Hong and Yang in -based spaces to classical functional settings in dispersive PDEs involving the smoothing effect and maximal function estimates, as originally introduced in the pioneering works of Kenig, Ponce and Vega. We believe that our approach may therefore be adapted to tackle continuum limits of more general dispersive equations.

Paper Structure

This paper contains 16 sections, 32 theorems, 210 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of results
  4. Outline
  5. Notation
  6. The linear equation
  7. Definitions on the lattice
  8. Discrete linear equation
  9. Filtering initial data
  10. Lwp of the discrete model
  11. Continuum limit
  12. General strategy
  13. Interpolation
  14. Comparison estimates
  15. Main argument
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\sigma=\frac{\alpha}{\beta}$, and suppose that where $\alpha\in (0,2)$ and $\beta\in (0,1)$. Then for every $f\in H^s(\mathbb R)$ there exists $T=T(\left\lVert f\right\rVert_{H^s(\mathbb R)})>0$ (with $T(\rho)\rightarrow\infty$ as $\rho\rightarrow 0$) and a unique solution $u_h(t)$ to the integral equation associated to eq:intro_discrete satisfy and Moreover, for $T'<T$, the map $f \mapsto

Figures (1)

  • Figure 1: Graph of $w(\xi)^{1/\beta}$ (blue) and $|\xi|^{\alpha/\beta}$ (red) for $\alpha=1.75$ and $\beta=0.8$ (left), and their derivatives (right). Produced with Wolfram Mathematica.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Continuum limit
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3: Discrete Sobolev inequality
  • ...and 52 more