Stability of traveling waves for the Burgers-Hilbert equation

Abstract

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

Some typos and minor errors fixed

References in corpus (3)

• The wave breaking for Whitham-type equations revisited
• Shock formation for the Burgers-Hilbert equation
• On the approximation of vorticity fronts by the Burgers-Hilbert equation