Solitary Water Waves with Ripples Beyond All Orders

Abstract

The famous solitary water wave, first observed and studied by Russell in the mid-nineteenth century, is best known in a mathematical context as a solution of the Korteweg-de Vries equation, which describes long water waves of small amplitude. Solitary waves have received lasting attention because of their occurrence in several important physical settings, their striking persistence, and the remarkable structure of the KdV and related equations which has emerged over the last 25 years. Our interest here is in the extent to which actual solutions of the equations of water waves correspond to the soliton solution of the KdV. For the case in which surface tension is neglected, it was shown in the 1940’s and 1950’s by Lavrentiev and by Friedrichs and Hyers that there are exact progressing waves whose first approximation are the solitons of the KdV equation. The case with surface tension included is genuinely different, however, because of a resonance with linear waves of the same speed. The solitary waves have speed slightly greater than C ₀ = √gh, where h is the depth of the water. Without surface tension, the linear waves of any wave number have speed < C ₀, so that there is no overlap in the speeds of the two families. When surface tension is introduced, the dispersion relation changes, and the speed crosses C ₀ as the wave number increases. Thus, in searching for a solitary wave of a certain speed, we may pick up the linear wave of the same speed. It appears that solitary waves can exist only in combination with the linear waves (called capillary waves or ripples), at least for most choices of the parameters.