The double cnoidal wave of the Korteweg–de Vries equation: An overview

Abstract

Earlier work of the author on the spatially periodic solutions of the Korteweg–de Vries equation is here extended via an in-depth treatment of a special case. The double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by Korteweg and de Vries in 1895. In the limit of small amplitude, the double cnoidal wave is the sum of two noninteracting linear sine waves. In the oppositie limit of large amplitude, it is the sum of solitary waves of two different heights repeated periodically over all space. Although special, the double cnoidal wave is important because it is but the particular case N=2 of a broad family of solutions known variously as ‘‘N-polycnoidal waves,’’ ‘‘finite gap,’’ ‘‘finite zone’’ solutions, ‘‘waves on a circle,’’ or ‘‘N-phase wave trains.’’ It has been shown by others that the set of N-polycnoidal waves gives the general initial value solution to the Korteweg–de Vries equation. This present work is the core of a three-part treatment of the double cnoidal wave. This part, the overview, presents graphic examples in all the important parameter regimes, explains how collision phase shifts alter the average speed of the two wave phases from the ‘‘free’’ velocities of the two solitary waves, describes the different branches or modes of the double cnoidal wave (it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes), and contrasts the results of this work with the very limited numerical calculations of previous authors. The second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory. The third part explains how the so-called ‘‘modular transformation’’ of the Riemann theta functions is important in interpreting N-polycnoidal waves.