Double cnoidal waves of the Korteweg-de Vries equation: A boundary value approach

Abstract

Double cnoidal waves of the Korteweg-de Vries equation are studied by direct solution of the nonlinear boundary value problems. These double cnoidal waves, which are the spatially periodic generalization of the well-known double soliton, are exact solutions with two independent phase speeds. The equation is written in terms of two phase variables and expanded in two-dimensional Fourier series. The small-amplitude solution is obtained via the Stokes' perturbation expansion. This solution is numerically extended to larger amplitude by employing a Newton-Kantorovich⧸continuation in amplitude⧸ Galerkin algorithm. The crests of the finite amplitude solution closely match the sech 2 solitary wave form and the three cases of solitary wave interaction described by Lax are identified for the double cnoidal waves. This simple approach reproduces specific features such as phase shift upon collision, distinction between instantaneous and average phase speeds, and a “paradox of wavenumbers”.