On numerical approaches to nonlinear Schrödinger and Korteweg–de Vries equations for piecewise smooth and slowly decaying initial data

Abstract

We discuss numerical approaches for nonlinear Schrödinger and Korteweg–de Vries equations with slowly decaying and piecewise smooth initial data. An efficient time integration for these equations is based on exponential integrators in the Fourier domain. The important step in such approaches is the computation of rapidly oscillating integrals on suitably chosen contours in the complex plane using spectral methods. Several examples mainly in the linear case are discussed.