A non-local approach to waves of maximal height for the Degasperis-Procesi equation

TL;DR

This paper analyzes the non-local formulation of the Degasperis-Procesi equation, showing that all bounded traveling-wave solutions are either smooth or peaked, with the highest solutions being peaked and no cuspon solutions existing.

Contribution

It characterizes the maximal height solutions of the Degasperis-Procesi equation and demonstrates the absence of cuspon solutions through a bifurcation analysis.

Findings

All bounded solutions are either smooth or peaked.

The highest solutions are peaked at their maximum height.

The equation does not admit cuspon solutions.

Abstract

We consider the non-local formulation of the Degasperis-Procesi equation $u_t+uu_x+L(\frac{3}{2}u^2)_x=0$, where $L$ is the non-local Fourier multiplier operator with symbol $m(\xi)=(1+\xi^2)^{-1}$. We show that all $L^\infty$, pointwise travelling-wave solutions are bounded above by the wave-speed and that if the maximal height is achieved they are peaked at those points, otherwise they are smooth. For sufficiently small periods we find the highest, peaked, travelling-wave solution as the limiting case at the end of the main bifurcation curve of $P$-periodic solutions. The results imply that the Degasperis-Procesi equation does not admit cuspon solutions.