Stability of traveling waves for the Burgers-Hilbert equation

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 . We use a modified energy method to prove the existence time of smooth solutions on a time scale of 1 δ with 0 < δ 1 and on a time scale of δ 2 with 0 < δ 2 1. Moreover, we show that the traveling wave exists for an amplitude in the range (0, * ) with * ∼ 0.29 and fails to exist for > 2 e .

Highest Cusped Waves for the Burgers-Hilbert equation

“…For u 0 the first Fourier coefficient is given by −0.54771699..., taking into account the bounds for v we can compute the enclosure [0.534, 0.561] for the first Fourier coefficient of ϕ. This agrees with the results from Castro, Córdoba and Zheng [10] that the branch of solutions break down for somewhere between * ∼ 0. 23 and 2 e .…”

“…Castro, Córdoba and Zheng [10] proved that this branch exists in the range (0, * ) with * ∼ 0.23 and fails to exist for > 2 e . Moreover they proved an enhanced lifespan estimate for perturbations of ϕ compared to the results in [26].…”

A nonlocal model describing tumor angiogenesis

“…There, the singularities occur [9,34] but they do at later times than suggested by standard energy estimates [19,20]. Also, stability of travelling waves [10] and global existence of weak solutions are known [5].…”