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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
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(𝜖δ), with 0 < δ 𝜖 1, and on a time scale of 𝜖δ2, with 0 < δ 𝜖2 1. Moreover, we show that the traveling wave exists for an amplitude 𝜖 in the range (0,𝜖), with 𝜖 0.23, and fails to exist for 𝜖 > 2e.

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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