Waves of maximal height for a class of nonlocal equations with homogeneous symbols

TL;DR

This paper investigates the existence, regularity, and properties of peaked periodic traveling-wave solutions in a class of nonlocal equations with homogeneous symbols, revealing the highest wave as a Lipschitz continuous limit.

Contribution

It introduces a bifurcation-based approach to identify the highest peaked wave in nonlocal equations with homogeneous symbols, including reformulating the Ostrovsky equation in a nonlocal form.

Findings

Existence of a highest peaked wave with Lipschitz regularity.

Application of bifurcation theory to nonlocal equations.

Recovery of the unique highest wave for the reduced Ostrovsky equation.

Abstract

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2\pi$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.