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

TL;DR

This paper investigates nonlocal equations with Bessel potential kernels, proving the existence of a highest, Lipschitz continuous, even, $2\\pi$-periodic traveling wave solution using bifurcation methods.

Contribution

It introduces a novel application of bifurcation techniques to establish the existence and regularity of maximal height waves in nonlocal equations with inhomogeneous symbols.

Findings

Existence of a highest, Lipschitz continuous, $2\pi$-periodic traveling wave.

Wave is even and has maximal height.

Regularity of the wave is exactly Lipschitz.

Abstract

In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order $\alpha$ for $\alpha > 1$. Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, $2\pi$-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz.