Linear instability of a Burgers--Hilbert traveling wave
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914531648733184 |
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| author | Castro, Ángel Gómez-Serrano, Javier Pascual-Caballo, Miguel M. G. |
| author_facet | Castro, Ángel Gómez-Serrano, Javier Pascual-Caballo, Miguel M. G. |
| contents | We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $ω$ and large wave speed $c$. For $ω= 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03920 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear instability of a Burgers--Hilbert traveling wave Castro, Ángel Gómez-Serrano, Javier Pascual-Caballo, Miguel M. G. Analysis of PDEs We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $ω$ and large wave speed $c$. For $ω= 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states. |
| title | Linear instability of a Burgers--Hilbert traveling wave |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.03920 |