Existence of All Wilton Ripples of the Kawahara Equation

Ryan P. Creedon

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arXiv · 2411.13508

Existence of All Wilton Ripples of the Kawahara Equation

Ryan P. Creedon

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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.

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Ryan P. Creedon. 2024-11-20. Existence of All Wilton Ripples of the Kawahara Equation. https://arxiv.org/abs/2411.13508

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