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The Connection of the Degasperis-Procesi equation with the Vakhnenko equation

Vakhnenko, V.O. and Parkes, J. (2004) The Connection of the Degasperis-Procesi equation with the Vakhnenko equation. In: Proceedings of Institute of Mathematics of NAS of Ukraine. Institute of Mathematics of NAS, pp. 493-497. ISBN 966-02-3224-1

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Travelling-wave solutions of the Degasperis-Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodicwave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well. Hone and Wang showed a connection between the DPE and the Vakhnenko equation (VE). Comparing the solutions of the DPE and the VE, we observe that, for both equations at interaction of waves, there are three kinds of phaseshift that depend on the ratio of wave amplitudes. In particular, there is a case when two interacted waves have phaseshifts in the positive direction.