Numerical Integrators for Dispersion-Managed KdV Equation

Abstract

In this paper, we consider the numerics of the dispersion-managed Korteweg-de Vries (DM-KdV) equation for describing wave propagations in inhomogeneous media. The DM-KdV equation contains a variable dispersion map with discontinuity, which makes the solution non-smooth in time. We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation, where a necessary constraint on the time step has been identified. Then, two exponential-type dispersion-map integrators up to second order accuracy are derived, which are efficiently incorporated with the Fourier pseudospectral discretization in space, and they can converge regardless of the discontinuity and the step size. Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast & strong dispersion-management regime.