Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

Abstract

This paper concerns the existence of multi-hump traveling waves propagating on the free surface of a two-dimensional water channel under the influence of gravity and small surface tension force. The fluid of constant density is assumed to be inviscid and incompressible and the flow is irrotational. It was known that the exact governing equations, called Euler equations, possess a generalized solitary-wave solution of elevation that consists of a single crest (or hump) at the center and a much smaller oscillation at infinity. This paper provides the first proof of the existence of multi-hump waves using the Euler equations. It is shown that when the wave speed is near its critical value and the surface tension is small, the Euler equations have a two-hump solution which consists of two crests, that are spaced far apart, and a smaller oscillation at infinity. Moreover, the ideas and methods may be used to study $$2^m$$-hump solutions.