Fourier Spectral Method for a Class of Nonlinear Schrödinger Models

Lei Zhang,Xuan Liu,Haidong Qu,Weihua Ou Yang,Jorge E Macias-Diaz

doi:10.1155/2021/9934858

Lei Zhang, Xuan Liu + Show 3 more

Open Access

https://doi.org/10.1155/2021/9934858

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Journal: Advances in Mathematical Physics	Publication Date: Jul 1, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Hanshan Normal University

Abstract

In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.

Highlights

Schrödinger equation, being known as basic assumption of quantum mechanics, is one of the most important equations in quantum mechanics that proposed by Austrian physicist Schrödinger
We study the numerical solution to the nonlinear Schrödinger equation of the following form: iutðx, tÞ + αΔuðx, tÞ + βjuðx, tÞj2uðx, tÞ + γf ðxÞuðx, tÞ = 0, ð1Þ
We prove that the semidiscrete Fourier spectral scheme (14) can keep the conservation laws

Summary

Introduction

Schrödinger equation, being known as basic assumption of quantum mechanics, is one of the most important equations in quantum mechanics that proposed by Austrian physicist Schrödinger. In literature [7], Wang et al proposed a linearly implicit conservative difference scheme for the coupled nonlinear Schrödinger equations with space fractional derivative, and she Advances in Mathematical Physics proved that the scheme can conserve the mass and energy in the discrete level. In 2015, Wang et al [8] studied the energy conservative Crank-Nicolson difference scheme for nonlinear Riesz space-fractional Schrödinger equations and gave the proof of mass conservation and energy conservation in the discrete sense. We study the numerical solution to the nonlinear Schrödinger equation of the following form: iutðx, tÞ + αΔuðx, tÞ + βjuðx, tÞj2uðx, tÞ + γf ðxÞuðx, tÞ = 0, ð1Þ with the periodic condition uðx + 2π, tÞ = uðx, tÞ, ð2Þ and the initial condition uðx, 0Þ = u0ðxÞ, ð3Þ where.

Preliminaries

The Full-Discrete Fourier Spectral Scheme and Its Conservation

Applying

Conclusion