Continuum Limit of 2D Fractional Nonlinear Schr\"odinger Equation

TL;DR

This paper demonstrates that solutions of a discrete 2D fractional nonlinear Schr"odinger equation converge to the continuum version as the discretization becomes finer, using dispersive estimates and asymptotic analysis.

Contribution

It establishes the continuum limit of the discrete fractional NLS with non-local coupling, providing uniform dispersive estimates and analyzing boundary behavior of constants.

Findings

Strong convergence in $L^2(\

Dispersive estimates are uniform in the discretization parameter.

Constants blow up as non-locality parameter approaches boundaries.

Abstract

We prove that the solutions to the discrete Nonlinear Schr\"odinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schr\"odinger Equation (FNLSE), as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $\alpha \in (1,2)$ approaches the boundaries.