A Jacobi-Galerkin Spectral Method for Computing the Ground and First Excited States of Nonlinear Fractional Schrödinger Equation

Abstract

The behaviour of the ground and first excited states of the nonlinear fractional Schrödinger equation is studied by an approximation method. In order to determine the nonlinear term of the problem under consideration, a normalised fractional gradient flow is introduced and the decay of a modified energy is established. The problem is then discretised by a semi-implicit Euler method in time and Jacobi-Galerkin spectral method in space. One- and two-dimensional numerical examples show that the strong nonlocal interactions lead to a large scattering of particles. Moreover, numerical simulations confirm the fundamental gap conjecture and show that for small interactions the ground and first excited states are more peaked and narrower.