A priori error analysis of linear and nonlinear periodic Schr{\"o}dinger equations with analytic potentials

TL;DR

This paper conducts an a priori error analysis for linear and nonlinear periodic Schrödinger equations with analytic potentials, examining how regularity affects solutions and the convergence of Fourier discretization methods.

Contribution

It provides new insights into the error behavior of Fourier methods for nonlinear Schrödinger equations with analytic potentials.

Findings

Regularity of potentials influences solution smoothness in linear cases.

Fourier discretization converges at specific rates depending on regularity.

Nonlinear cases exhibit different regularity transfer compared to linear cases.

Abstract

This paper is concerned with the numerical analysis of linear and nonlinear Schr{\"o}dinger equations with analytic potentials. While the regularity of the potential (and the source term when there is one) automatically conveys to the solution in the linear cases, this is no longer true in general in the nonlinear case. We also study the rate of convergence of the planewave (Fourier) discretization method for computing numerical approximations of the solution.