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 , where L is the non-local Fourier multiplier operator with symbol . We show that all , 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 there are no travelling cuspon solutions to the Degasperis-Procesi equation.