Convergence analysis of time-splitting projection method for nonlinear quasiperiodic Schr\"odinger equation

TL;DR

This paper introduces and analyzes a numerical method combining space projection and Strang splitting to efficiently solve nonlinear quasiperiodic Schrödinger equations, proving spectral accuracy in space and second-order accuracy in time.

Contribution

It develops a novel analysis framework for the convergence of a space-time splitting method tailored to quasiperiodic Schrödinger equations, addressing unique challenges of quasiperiodic structures.

Findings

Spectral accuracy achieved in space

Second-order accuracy in time

Numerical experiments confirm theoretical results

Abstract

This work proposes and analyzes an efficient numerical method for solving the nonlinear Schr\"odinger equation with quasiperiodic potential, where the projection method is applied in space to account for the quasiperiodic structure and the Strang splitting method is used in time.While the transfer between spaces of low-dimensional quasiperiodic and high-dimensional periodic functions and its coupling with the nonlinearity of the operator splitting scheme make the analysis more challenging. Meanwhile, compared to conventional numerical analysis of periodic Schr\"odinger systems, many of the tools and theories are not applicable to the quasiperiodic case. We address these issues to prove the spectral accuracy in space and the second-order accuracy in time. Numerical experiments are performed to substantiate the theoretical findings.