A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations

Deng, Xijun

doi:10.1016/j.amc.2011.07.044

Applied Mathematics and Computation

2011

DOI: 10.1016/j.amc.2011.07.044

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A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations

Xijun Deng

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New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

Xiang

Davodi

Mirhosseini-Alizamini

et al. 2021

Mod. Phys. Lett. B

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Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated [Formula: see text]-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the [Formula: see text]-fractional heat equation is found. Graphical representations are also given.

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New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

Xiang

Davodi

Mirhosseini-Alizamini

et al. 2021

Mod. Phys. Lett. B

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Multi-shockpeakons for the stochastic Degasperis-Procesi equation

Arruda

2022

era

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<abstract>The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u(x, t) = \sum\limits_{i = 1}^nm_i(t)e^{-|x-x_i(t)|}-\sum\limits_{i = 1}^ns_i(t){\rm sgn}(x-x_i(t))e^{-|x-x_i(t)|}, $\end{document} </tex-math></disp-formula> where $ {\rm sgn}(x) $ denotes the signum function with $ {\rm sgn}(0) = 0 $, if and only if the time-dependent parameters $ x_i(t) $ (positions), $ m_i(t) $ (momenta) and $ s_i(t) $ (shock strengths) satisfy a system of $ 3n $ ordinary differential equations. We prove that a stochastic perturbation of the Degasperis-Procesi equation also has weak multi-shockpeakon solutions if and only if the positions, momenta and shock strengths obey a system of $ 3n $ stochastic differential equations.</abstract>

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A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations

Cited by 2 publications

References 20 publications

New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

Multi-shockpeakons for the stochastic Degasperis-Procesi equation

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