A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schr\"odinger equation

TL;DR

This paper introduces a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation that is linearly implicit, invariant-preserving, and efficiently solvable without step size restrictions.

Contribution

It presents a novel linearly implicit, structure-preserving spectral scheme with guaranteed solvability and efficient linear system solutions for the space-fractional nonlinear Schrödinger equation.

Findings

Scheme preserves two invariants of the equation

Unique solvability without restrictions on step sizes

Efficient solution via variable transformation and preconditioning

Abstract

We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schr\"odinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.