Wilton Ripples in Weakly Nonlinear Dispersive Models of Water Waves: Existence and Analyticity of Solution Branches

Abstract

Traveling waves on the surface of the ocean play an important role in many oceanographic processes which necessitates a detailed quantitative understanding of their properties. The water wave equations, which govern the free-surface evolution of an ideal fluid, are the most successful model for this phenomena, but are exceedingly difficult to analyze due to their strongly nonlinear character and the fact that they are posed on a domain with moving boundary. For this reason, weakly nonlinear dispersive models are an essential tool for practitioners, and in this contribution, we study traveling wave solutions of a broad class of such models. The simplest family of traveling wave solutions are the Stokes waves which can be characterized as simple bifurcation (one-dimensional null space of the linearized operator) from the trivial (flat-water) branch of solutions. We focus our analysis on the much less studied non-simple case of Wilton ripples which have linear behavior characterized by two co-propagating harmonics (a two-dimensional null space of the linearized operator). More specifically, we show that such branches of solutions exist for a class of nonlinear dispersive model equations, and that they are analytic with respect to a natural wave height/slope parameter.