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

Abstract

We consider the non-local formulation of the Degasperis-Procesi equation $u_t+u{u}_x+L(\frac{3}{2}{u}^2{)}_x=0$, 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.