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arXiv · 2605.03920

Linear instability of a Burgers--Hilbert traveling wave

Ángel Castro·Javier Gómez-Serrano·Miguel M. G. Pascual-Caballo
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Abstract

We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $\omega$ and large wave speed $c$. For $\omega = 3$ 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.

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Ángel Castro, Javier Gómez-Serrano, Miguel M. G. Pascual-Caballo. 2026-05-05. Linear instability of a Burgers--Hilbert traveling wave. https://arxiv.org/abs/2605.03920

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