32 - Hyperfast Modeling of Shallow-Water Waves: The KdV and KP Equations

Abstract

This chapter discusses the hyperfast modeling of shallow-water waves with the help of Korteweg-de Vries (KdV) and Kadomtsev–Petviashvili (KP) equations. It also discusses the inverse scattering transform (IST), with periodic boundary conditions, for the KdV and KP equations and the nonlinear spectral representation for shallow-water wave trains in terms of the Riemann (spectral) matrix and a set of arbitrary (possibly random) phases. Important useful properties of Riemann theta functions that lead to the new numerical algorithm and how to write the theta function as a two-dimensional, ordinary linear Fourier series with time varying coefficients that have the form of temporal Fourier series with amplitudes that are a function of the Riemann matrix and with incommensurable frequencies are discussed. The Schottky uniformization procedure that helps to easily characterize the Riemann spectrum in terms of the physically and numerically convenient uniformization parameters is presented. A procedure is given for computing the linear Fourier components for the wave-field solution of the KP equation in terms of IST parameters. The procedure for a fast computation of Riemann theta functions is described, and the details for the computation of nonlinear, directional KP wave trains from directional spectra are given.