Growth of Sobolev norms and strong convergence for the discrete nonlinear Schr{\"o}dinger equation

TL;DR

This paper proves that solutions of the discrete nonlinear Schrödinger equation strongly converge to those of the continuous version in Sobolev norms, specifically in 1D and 2D, using bilinear estimates and growth control techniques.

Contribution

It establishes strong Sobolev norm convergence of discrete NLS solutions to continuous solutions, with new bilinear estimates and growth control methods.

Findings

Strong convergence in Sobolev norms for 1D and 2D cases

Development of bilinear estimates for Shannon interpolation

Control of discrete Sobolev norm growth

Abstract

We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to the one and two-dimensional case, with a set of parameters which implies global well-posedness for the continuous equation. Our proof relies on the use of bilinear estimates for the Shannon interpolation as well as the control of the growth of discrete Sobolev norms that we both prove.