John PARKES
University of Strathclyde, Richmond St.,
Glasgow G1 1XH, UK
E-mail: ejp@maths.strath.ac.uk

Connection of Degasperis-Procesi equation with Vakhnenko equation

Abstract:
Travelling-wave solutions of the Degasperis-Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodic-wave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well.

Hone and Wang showed the connection of the DPE with the Vakhnenko equation (VE).  Recently we have studied the VE and have found two families of solutions. In one case, there are loop-like solitary waves and periodic-loop solutions; in other case, there are periodic-hump and periodic-spike solutions.

Comparing the solutions of the DPE and the VE, we observe that for the two equations at interaction of the waves, there are three kinds of the phaseshift in dependence on ratio of wave amplitudes. In particular, there is a case, when the two interacted waves have the phaseshifts in positive direction. For soliton of the VE this phenomena can be explained that the soliton "mass" is equal to zero.