Linear instability of a Burgers--Hilbert traveling wave

TL;DR

This paper demonstrates spectral instability of certain traveling wave solutions to the Burgers--Hilbert equation using computer-assisted methods, indicating potential instability in related vortex patch configurations.

Contribution

It introduces a computer-assisted approach to analyze spectral stability of Burgers--Hilbert traveling waves, revealing instability for specific parameters.

Findings

Eigenvalue with negative real part at specified parameters

Spectral instability of the solutions

Implications for vortex patch stability in Euler equations

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.