Continuum Limit for Discrete NLS with Memory Effect (Journal of Nonlinear Modeling and Analysis, 2025)
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 transportin biomolecules. Because the distance between base pairs is small, we considerthe continuum limit: a sharp approximation of the system as $h → 0.$ In thislimit, 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 theseresults, we generalize some recent ideas proposed by Hong and Yang in $L^2$-based spaces to classical functional settings in dispersive PDEs involving thesmoothing effect and maximal function estimates, as originally introduced inthe pioneering works of Kenig, Ponce and Vega. We believe that our approachmay therefore be adapted to tackle continuum limits of more general dispersiveequations.