Highest waves for fractional Korteweg--De Vries and Degasperis--Procesi equations

TL;DR

This paper investigates traveling wave solutions for fractional Korteweg--De Vries and Degasperis--Procesi equations, revealing the existence of smooth periodic waves and a highest cusped wave with optimal regularity.

Contribution

It establishes the existence of local bifurcation branches of traveling waves and characterizes the highest cusped wave solution for fractional equations.

Findings

Existence of smooth, periodic traveling waves from bifurcation analysis.

Identification of a highest cusped wave with optimal regularity.

Extension of local solutions to global solution curves.

Abstract

We study traveling waves for a class of fractional Korteweg--De Vries and fractional Degasperis--Procesi equations with a parametrized Fourier multiplier operator of order $-s \in (-1, 0)$. For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal $s$-H\"older regularity, attained in the cusp.