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

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 L² norm for H¹ initial data. A spectral method in space, using fast Fourier transforms with 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 L²-norm error bound of O(τ[ln(1/τ)]² + N⁻¹) for H¹ initial data, where τ is the time-step size.