2019/10/13 by Ricardo Grande, Grande, Ricardo
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1910.05681
openalex publication_date 2019/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ 0. In this limit, we prove that solutions to this discrete equation converge strongly in L2 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 L2-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.