Stability of traveling waves for the Burgers–Hilbert equation

Castro, Ángel; Córdoba, Diego; Zheng, Fan

doi:10.2140/apde.2023.16.2109

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Stability of traveling waves for the Burgers–Hilbert equation

Ángel Castro, Diego Córdoba and Fan Zheng

Vol. 16 (2023), No. 9, 2109–2145

DOI: 10.2140/apde.2023.16.2109

Abstract

We consider smooth solutions of the Burgers–Hilbert equation that are a small perturbation $δ$ from a global periodic traveling wave with small amplitude $𝜖$ . We use a modified energy method to prove the existence time of smooth solutions on a time scale of $1 ∕ (𝜖 δ)$ , with $0 < δ ≪ 𝜖 ≪ 1$ , and on a time scale of $𝜖 ∕ δ^{2}$ , with $0 < δ ≪ 𝜖^{2} ≪ 1$ . Moreover, we show that the traveling wave exists for an amplitude $𝜖$ in the range $(0, 𝜖^{*})$ , with $𝜖^{*} \sim 0.23$ , and fails to exist for $𝜖 > 2 ∕ e$ .

Keywords

Burgers–Hilbert, normal forms, traveling waves

Mathematical Subject Classification

Primary: 35F25, 76B47

Milestones

Received: 3 March 2021

Revised: 22 March 2022

Accepted: 29 April 2022

Published: 11 November 2023

Authors

Ángel Castro
Instituto de Ciencias Matemáticas ICMAT-CSIC-UAM-UCM-UC3M Madrid Spain

Diego Córdoba
Instituto de Ciencias Matemáticas ICMAT-CSIC-UAM-UCM-UC3M Madrid Spain

Fan Zheng
Instituto de Ciencias Matemáticas ICMAT-CSIC-UAM-UCM-UC3M Madrid Spain

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Vol. 16, No. 9, 2023