Numerical simulation of a modified KdV equation on the whole real axis

Abstract

The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme.