A constructive low-regularity integrator for the one-dimensional cubic nonlinear Schrödinger equation under Neumann boundary condition

Abstract

Abstract A new harmonic analysis technique using the Littlewood–Paley dyadic decomposition is developed for constructing low-regularity integrators for the one-dimensional cubic nonlinear Schrödinger equation in a bounded domain under Neumann boundary condition, when the frequency analysis based on the Fourier series cannot be used. In particular, a low-regularity integrator is constructively designed through the consistency analysis by the Littlewood–Paley decomposition of the solution, in order to have almost first-order convergence (up to a logarithmic factor) in the $L^{2}$ norm for $H^{1}$ initial data. A spectral method in space, using fast Fourier transforms with $\mathcal{O}(N\ln N)$ operations at every time level, is constructed without requiring any Courant-Friedrichs-Lewy (CFL) condition, where $N$ is the degrees of freedom in the spatial discretization. The proposed fully discrete method is proved to have an $L^{2}$-norm error bound of $\mathcal{O}(\tau [\ln (1/\tau )]^{2}+ N^{-1})$ for $H^{1}$ initial data, where $\tau $ is the time-step size.