Numerical study of a nonlocal nonlinear Schr\"odinger equation (MMT model)

TL;DR

This paper numerically investigates a nonlocal nonlinear Schr"odinger equation (MMT model), analyzing solution behavior, stability of standing waves, and semi-classical limits in both focusing and defocusing regimes.

Contribution

It introduces numerical methods to generate and analyze the stability of standing wave solutions for the nonlocal NLS equation, and explores long-term dynamics and semi-classical limits.

Findings

Standing wave solutions are numerically generated and their stability analyzed.

Conditions for global boundedness of solutions are identified.

Semi-classical limits are explored in both focusing and defocusing cases.

Abstract

In this paper, we study a nonlocal nonlinear Schr\"odinger equation (MMT model). We investigate the effect of the nonlocal operator appearing in the nonlinearity on the long-term behavior of solutions, and we identify the conditions under which the solutions of the Cauchy problem associated with this equation is bounded globally in time in the energy space. We also explore the dynamical behavior of standing wave solutions. Therefore, we first numerically generate standing wave solutions of nonlocal nonlinear Schr\"odinger equation by using the Petviashvili's iteration method and their stability is investigated by the split-step Fourier method. This equation also has a two-parameter family of standing wave solutions. In a second step, we meticulously concern with the construction and stability of a two-parameter family of standing wave solutions numerically. Finally, we investigate the semi-classical limit of the nonlocal nonlinear Schr\"odinger equation in both focusing and defocusing cases.