High-order numerical method for scattering data of the Korteweg—De Vries equation

Abstract

Nonlinear wavefields governed by integrable models such as the Korteweg–De Vries (KdV) equation can be decomposed into the so-called scattering data playing the role of independent elementary harmonics evolving trivially in time. A typical scattering data portrait of a spatially localised wavefield represents nonlinear coherent wave structures (solitons) and incoherent radiation. In this work we present a fourth-order accurate algorithm to compute the scattering data within the KdV model. The method based on the Magnus expansion technique provides accurate information about soliton amplitudes, velocities and intensity of the radiation. Our tests performed using a box-shaped wavefield confirm that all components of the scattering data are computed correctly, while the test based on a single-soliton solution verifies the declared order of a numerical scheme.