Continuum limit of 3D fractional nonlinear Schr\"odinger equation

TL;DR

This paper proves that solutions of the discrete 3D fractional nonlinear Schrödinger equation converge to the continuous version as the lattice spacing approaches zero, using uniform Strichartz estimates and advanced oscillatory integral techniques.

Contribution

It establishes the continuum limit of the 3D fractional nonlinear Schrödinger equation with a novel uniform Strichartz estimate for the discrete case.

Findings

Strong convergence of discrete to continuous solutions as h->0

Development of uniform Strichartz estimates for discrete fractional Schrödinger equations

Application of oscillatory integral and Newton polyhedron techniques

Abstract

In this paper, we investigate the continuum limit theory of the fractional nonlinear Schr\"odinger equation in dimension 3. We show that the solution of discrete fractional nonlinear Schr\"odinger equation on hZ^3 will converge strongly in L^2 to the solution of fractional nonlinear Schr\"odinger equation on R^3, when h->0. The key is proving the uniform-in-h Strichartz estimate for discrete fractional nonlinear Schr\"odinger equation, by using the uniform estimate of oscillatory integral and Newton polyhedron techniques.