A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

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.


Introduction
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 [1] [2], the homotopy analysis method [3] [4], the variational iteration method [5] [6], the homotopy perturbation method [7]- [10], and variational homotopy perturbation method [11] [12] 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. Wazwaz [13] studies a family of important physically equations which is called modified b -equation. It has the following expression:

Mathematical Models
where b is a positive integer. As is known, when 2, The mDP equation with exact solution [13]:

Variational Iteration Method (VIM)
To clarify the basic ideas of VIM, we consider the following differential equation where L is a linear operator defined by , where λ is a general lagrangian multiplier by [14] defined as: The subscript n indicates the nth approximation and n u  is considered as a restricted variation [15].

Homotopy Perturbation Method (HPM)
To illustrate the basic idea of this method, we consider the following nonlinear differential equation: with the boundary condition where A is a general differential operator, B a boundary operator, ( ) f r a known analytical function and Γ is the boundary of the domain Ω . A can be divided into two parts which are L and N , where L is linear and N is nonlinear. Equation (9) can therefore be rewritten as follows: Homotopy perturbation structure is shown as follows: In Equation (12), is an embedding parameter and 0 u is the first approximation that satisfies the boundary condition. We can assume that the solution of Equation (12) can be written as a power series in p , as following: and the best approximation for solution is: It is well known that series (15) is convergent for most of the cases and also the rate of convergence depends on ( ) L u . We assume that Equation (15) has a unique solution [7].

Variational Homotopy Perturbation Method (VHPM)
To illustrate the concept of the variational homotopy perturbation method, we consider the general differential Equation (6). We construct the correction functional (7) and apply the homotopy perturbation method (14) to obtain: As we see, the procedure is formulated by the coupling of variational iteration method and homotopy perturbation method. A comparison of like powers of p gives solutions of various orders.

Application of VHPM
In this section, we apply the variational homotopy perturbation method to solve mCH and mDP equations.

Application of VHPM to Modified Camassa-Holm Equation
Consider the mCH equation To solve Equation (17), using VIM, we have the correction functional as: where n u  is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as Equation (8) is 1 λ = − , which yields the following iteration formula: Comparing the coefficient of like powers of p , we have ( ) For an arbitrary ( )( ) , 2,3, j u x t j =  we can use symbolic software programme such as Mathematica to calculate it in the same manner.
If only the two-term approximation of Equation (15) is sufficient, then the approximate solution of Equation (17) will be expressed as: From its expression one can see that it is also a solitary wave solution. Remark 1. It should be remarked that the graph drawn here and approximate solution using VHPM is in excellent agreement with HPM [16] and VIM [17].

Application of VHPM to Modified Degasperis-Procesi
Consider the mDP equation 2 4 3 0, To solve Equation (29), using VIM, we have the correction functional as: where n u  is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be determined as Equation (8)