Periodic and solitary-wave solutions of the Degasperis-Procesi equation.

*(English)*Zbl 1049.35162Summary: Travelling-wave solutions of the Degasperis-Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive \(x\)-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative \(x\)-direction, there are solutions which are just the mirror image in the \(x\)-axis of the aforementioned solutions. A transformed version of the Degasperis-Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive \(x\)-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative \(x\)-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.

##### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

35B10 | Periodic solutions to PDEs |

35Q51 | Soliton equations |

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\textit{V. O. Vakhnenko} and \textit{E. J. Parkes}, Chaos Solitons Fractals 20, No. 5, 1059--1073 (2004; Zbl 1049.35162)

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##### References:

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