Download this article
 Download this article For screen
For printing
Recent Issues

Volume 16
Issue 9, 1989–2240
Issue 8, 1745–1988
Issue 7, 1485–1744
Issue 6, 1289–1483
Issue 5, 1089–1288
Issue 4, 891–1088
Issue 3, 613–890
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
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

Open Access made possible by participating institutions via Subscribe to Open.