Highest Cusped Waves for the Burgers-Hilbert equation

TL;DR

This paper proves the existence of a highest cusped traveling wave solution for the Burgers-Hilbert equation, analyzing its asymptotic behavior using a combination of asymptotic analysis and computer assistance.

Contribution

It establishes the existence of a novel highest cusped wave solution for the Burgers-Hilbert equation with detailed asymptotic characterization.

Findings

Existence of a highest cusped traveling wave proven.

Asymptotic behavior at zero characterized.

Combination of analytical and computational methods used.

Abstract

In this paper we prove the existence of a periodic highest, cusped, traveling wave solution for the Burgers-Hilbert equation $f_{t} + f f_{x} = H[f]$ and give its asymptotic behaviour at $0$. The proof combines careful asymptotic analysis and a computer-assisted approach.