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Convergence analysis of time-splitting projection method for nonlinear quasiperiodic Schrödinger equation

Kai Jiang, Shifeng Li, Xiangcheng Zheng

Abstract

This work proposes and analyzes an efficient numerical method for solving the nonlinear Schrödinger 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 Schrödinger 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.

Convergence analysis of time-splitting projection method for nonlinear quasiperiodic Schrödinger equation

Abstract

This work proposes and analyzes an efficient numerical method for solving the nonlinear Schrödinger 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 Schrödinger 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.

Paper Structure

This paper contains 17 sections, 17 theorems, 108 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Function spaces
  5. Basic properties of quasiperiodic functions
  6. Numerical scheme and error estimate
  7. Strang splitting method
  8. Full discrete scheme
  9. Main result
  10. Auxiliary estimates
  11. Estimates in time
  12. Estimates in space
  13. Estimate of intermediate solutions
  14. Numerical implementation
  15. One-dimensional case
  16. ...and 2 more sections

Key Result

Proposition 1

For quasiperiodic functions $\phi,\varphi\in \hbox{QP}(\mathbb{R}^d)$, $\phi_p$ and $\varphi_p$ are the $n_1$- and $n_2$-dimensional parent functions of $\phi$ and $\varphi$, respectively. Assume that there exist projection matrices such that $\phi(\bm{x})=\phi_p(\bm{P}_1^T\bm{x})$ and $\varphi(\bm{x})=\varphi_p(\bm{P}_2^T\bm{x})$. Then for the projection matrix $\bm{P}=(\gamma_1,\cdots,\gamma_{n

Figures (2)

  • Figure 1: The soliton at different time $T$.
  • Figure 2: Fundamental soliton located at the center.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 22 more