Numerical study of fractional Nonlinear Schr\"odinger equations

TL;DR

This paper numerically investigates fractional nonlinear Schrödinger equations using Fourier spectral methods, exploring blow-up phenomena, stability of ground states, and long-term dynamics across various regimes in one dimension.

Contribution

It provides a detailed numerical analysis of fractional nonlinear Schrödinger equations, including ground state construction and dynamics, which was not comprehensively studied before.

Findings

Identification of conditions for finite time blow-up and global existence.

Numerical construction of ground state solutions.

Insights into stability and long-term behavior of solutions.

Abstract

Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schr\"odinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be computed in one spatial dimension, only. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states, and the long time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions to the fractional nonlinear Schr\"odinger equation.