Existence of All Wilton Ripples of the Kawahara Equation

Abstract

We investigate the existence of Wilton ripple solutions of the Kawahara equation. Without loss of generality, these are $2\pi$-periodic, traveling-wave solutions whose profiles at zero amplitude have a codimension-1 bifurcation from a linear combination of $\cos(x)$ and $\cos(Kx)$ for $K \in \mathbb{N} \setminus \{1\}$. Using a Lyapunov-Schmidt reduction, we prove the existence of these solutions for all $K$, in contrast to previous work demonstrating existence only for $K = 2$. Although the proof holds only for the Kawahara equation, many ideas introduced in the proof can be applied to more general contexts, including Wilton ripples of the gravity-capillary water wave equations.