Continuum limit for discrete NLS with memory effect

Abstract

We consider a discrete nonlinear Schrödinger equation with long-range interactions and a memory effect on the infinite lattice $h\mathbb{Z}$ 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 $h\to 0$. 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.

References in corpus (7)

• Fractional Schrodinger equation
• Non-diffusive transport in plasma turbulence: a fractional diffusion approach
• Fractional Cauchy problems on bounded domains
• Models for energy and charge transport and storage in biomolecules
• Continuum limit for the Ablowitz--Ladik system
• Space-time fractional NLS equation
• Continuum limit of the discrete nonlinear Klein-Gordon equation

Cited by in corpus (4)

• Global well-posedness for the cubic fractional NLS on the unit disk
• Global well-posedness and long-time behavior of the fractional NLS
• Growth of Sobolev norms and strong convergence for the discrete nonlinear Schr{ö}dinger equation
• On the continuum limit for the discrete Nonlinear Schrödinger equation on a large finite cubic lattice