Numerical study of fractional nonlinear Schrödinger equations.

Christian Klein,Christof Sparber,Peter Markowich

doi:10.1098/rspa.2014.0364

Abstract

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. 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 of the fractional nonlinear Schrödinger equation.

Highlights

By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. 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
This work is concerned with a numerical study for non-local dispersive equations of nonlinear fractional Schrödinger type
We show the solution of a defocusing fractional Schrödinger type (fNLS) equation (1.9) with s = 0.9 and initial data

Summary

Introduction

This work is concerned with a numerical study for non-local dispersive equations of nonlinear fractional Schrödinger type (fNLS). All of the above considerations paint a picture in which the theory for fNLS seems to follow closely the usual NLS results While this is certainly true for basic questions such as existence and uniqueness versus finite time blow-up, the non-local nature of (1.1) with s < 1 is expected to have a considerable influence on more qualitative properties of the solution. Numerical simulations are performed in order to study the influence of a non-local dispersion term on different mathematical questions, including: the particular type of finite time blow-up (e.g. self-similar or not), qualitative features of the associated ground states solutions (including their stability) and the possibility of well-posedness in the energy supercritical regime. We have conducted several numerical experiments on the stability of fractional ground states, which are summarized in appendix A

Numerical methods

Methods for the numerical study of blow-up

Numerical studies of finite time blow-up

The energy critical and supercritical regime

Conclusion