Existence of All Wilton Ripples of the Kawahara Equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908877292830720 |
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| author | Creedon, Ryan P. |
| author_facet | Creedon, Ryan P. |
| contents | We investigate the existence of Wilton ripple solutions of the Kawahara equation. Without loss of generality, these are $2π$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of All Wilton Ripples of the Kawahara Equation Creedon, Ryan P. Analysis of PDEs Dynamical Systems We investigate the existence of Wilton ripple solutions of the Kawahara equation. Without loss of generality, these are $2π$-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. |
| title | Existence of All Wilton Ripples of the Kawahara Equation |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2411.13508 |