Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation

In this contribution, we prove that small amplitude, resonant harmonic, spatially periodic traveling waves (Wilton ripples) exist in a family of weakly nonlinear PDEs which model water waves. The proof is inspired by that of Reeder and Shinbrot (Arch. Rat. Mech. Anal. 77:321–347, 1981) and complements the authors’ recent, independent result proven by a perturbative technique (Akers and Nicholls 2021). The method is based on a Banach Fixed Point Iteration and, in addition to proving that this iteration has Wilton ripples as a fixed point, we use it as a numerical method for simulating these solutions. The output of this numerical scheme and its performance are evaluated against a quasi-Newton iteration.