Stability of traveling waves for the Burgers–Hilbert equation

We consider smooth solutions of the Burgers–Hilbert equation that are a small perturbation $\delta$ 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 $1∕(𝜖\delta )$, with $0<\delta \ll 𝜖\ll 1$, and on a time scale of $𝜖∕{\delta }^2$, with $0<\delta \ll {𝜖}^2\ll 1$. Moreover, we show that the traveling wave exists for an amplitude $𝜖$ in the range $(0,{𝜖}^{\ast })$, with ${𝜖}^{\ast }\sim 0.23$, and fails to exist for $𝜖>2∕e$.