Peakons and coshoidal waves: Traveling wave solutions of the Camassa-Holm equation

Abstract

Camassa and Holm [1] have recently derived a new integrable wave equation. For the special case κ = 0, they showed it has solitary waves of the form c exp(−|; x − ct|;) which they named “peakons”. In this work, we derive a perturbation series for general κ which converges even at the peakon limit. We also give three analytical representations for the spatially periodic generalization of the peakon, the “coshoidal wave”. The three representations are (i) a closed form, analytical solution, (ii) a Fourier series with coefficients that are explicit rational functions, and (iii) an imbricate Fourier series, which is the superposition of an infinite number of peakons, each separated from its neighbors by distance P where P is the spatial period. Lastly, we have numerically tested the soliton superposition principle. Although the Camassa-Holm equation is integrable for general κ, it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases κ = 0, κ/c = 1 2 . However, the imbricate-soliton series is a very good approximate solution for general κ, even when the spatial period is small and the solution resembles a sine wave more than a solitary wave.