Author(s)： Majeed A. Yousif, Bewar A. Mahmood, Fadhil H. Easif
In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.
Many varieties of physical, chemical, and biological phenomena can be expressed in terms of nonlinear partial differential equations. In most cases, it is difficult to obtain the exact solution for these equations. Therefore analytical methods have been used to find approximate solutions. In recent years, many analytical methods such as the Adomian decomposition method  , the homotopy analysis method  , the variational iteration method  , the homotopy perturbation method -, and variational homotopy perturbation method   have been utilized to solve linear and nonlinear equations.
In this paper, we will use variational homotopy perturbation method to study the Modified Camassa-Holm and Degasperis-Procesi equations and obtain their analytical solutions.
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