Construction of exact solutions to the modified forms of DP and CH equations by analytical methods

In this work, we establish the exact solutions to the modified forms of Degasperis–Procesi (DP) and Camassa– Holm (CH) equations. The generalized (G’/G)-expansion and generalized tanh-coth methods were used to construct solitary wave solutions of nonlinear evolution equations. The generalized (G’/G)-expansion method presents a wider applicability for handling nonlinear wave equations. It is shown that the (G’/G)-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.


Introduction
In the recent years, the investigation of the traveling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena.During the past decades, both mathematicians and physicists have devoted considerable effort to the study of exact and numerical solutions of the nonlinear ordinary or partial differential equations corresponding to the nonlinear problems.Many powerful methods have been presented.For instance, Hirota's bilinear method [1], the inverse scattering transform [2], F-expansion method [3], sine-cosine method [4], homotopy perturbation method [5], homotopy analysis method [6,7], variational iteration method [9], tanh-coth method [10,11], Exp-function method [12,13,14,15], central difference and Newton iteration method [16], septic B-spline collocation method [17] and so on.Here, we use of an effective method, (G'/G)-expansion method, for constructing a range of exact solutions for the nonlinear partial differential equations, first proposed by Wang et al [19].Zhang et al. [20] examined the generalized (G'/G)-expansion method and its applications.Authors of [21] obtained the exact solutions for the symmetric regularized long wave equation using the (G'/G)-expansion method.Fazli and Manafian [22] applied the (G'/G)-expansion method for solving the couple Boiti-Leon-Pempinelli system.Also, Bekir [23] used to application of the (G'/G)-expansion method for nonlinear evolution equations.In [24], solitary wave and periodic wave solutions via (G'/G)-expansion method The CH equation has peaked solitary wave solutions of the form [40] u(x, t) where c is the wave speed.The name 'peakon', that is, solitary wave with slope discontinuities, was used to single them from general solitary wave solutions since they have a corner at the peak of height c [40].In this article an application of the proposed method to the modified forms of Degasperis-Procesi and Camassa-Holm equations is illustrated.we will investigate modified forms of the DP and the CH equations given by and respectively.In [25] Mustafa studied DP equation.Multi-peakon solutions of DP equation was investigated by Lundmark [26].Shen have obtained the new integrable equation with peakon and compactons [27].Chen have acquired the new type of bounded waves for DP equation [28].In [29,30] have obtained the integrable shallow water equation and completely integrable shallow water equation respectively.In [31,32] Liu and et al. have studied the CH equation and have obtained peaked wave solutions [31] and peakons [32].Peakons and periodic cusp waves of the generalized CH equation have been studied by [33].Also in [34] Tian and et al. have studied the generalized CH equation and obtained new peaked solitary wave solutions.Boyd [35] derived a perturbation series which converges even at the peakon limit, and gave three analytical representation for the spatially periodic generalization of the peakon, called "coshoidal wave".Cooper and Shepard [36] derived approximate solitary wave solution by using some variational functions.Constantin [37] gave a mathematical description of the existence of interacting solitary waves.Wazwaz [38] derived a solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations.A class of nonlinear fourth order variant of a generalized Camassa-Holm equation was investigated by [39] where have obtained compact and noncompact solutions.Recently, CH equation and some its generalized forms have been studied by many authors, for instance, Wazwaz [40] have obtained new solitary wave solutions to the modified forms of DP and CH equations.He [41] derived exact travelling wave solutions of a generalized CH equation using the integral bifurcation method.In [42] an integrable shallow water equation has been discussed with linear and nonlinear dispersion.Also, new integrable equation with peakon solutions has been studied by Degasperis [43].Explicit solutions of the Camassa-Holm equation have been obtained in [44].Guo [45] obtained periodic cusp wave solutions and single-solitons for the b-equation.Bäcklund transformation has been applied for the modified DGH equation by [46].In [47] bifurcations of travelling wave solutions for a variant of CH equation investigated by He.In [48] Rui et.al, applied the integral bifurcation method and its application for solving a family of third-order dispersive PDEs.Finally, Liu and Qian [49] have derived peakons and their bifurcation for the generalized CH equation.The article is organized as follows: In Section 2, first we briefly give the steps of the methods and apply these methods to solve the nonlinear partial differential equations.In Section 3, the application of the (G'/G)-expansion method to the modified DP equation will be introduced briefly.Also, Section 4 by using the results obtained in Section 2, the corresponding solutions of the modified CH equation can be obtained.Also a conclusion is given in Section 5. Finally some references are given at the end of this paper.

The ( G ′
G )-expansion method Step 1.For a given NLPDE with independent variables X = (x, t) and dependent variable u: can be converted to an ODE by transformation ξ = x − ct is wave variable.Also, c is constant to be determined later.
Step 2. We seek its solutions in the more general polynomial form as following where G(ξ) satisfies the second order LODE in the form where a 0 , a k (k = 1, 2, ..., m), λ and µ are constants to be determined later, a m = 0, but the degree of which is generally equal to or less than m − 1, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. ( 7).
Step 4. Solving the algebraic equations in Step 3, then substituting a i , ..., a m , c and general solutions of Eq. ( 9) into Eq.( 8) we can obtain a series of fundamental solutions of Eq. ( 6) depending of the solution G(ξ) of Eq. ( 9).

The generalized tanh-coth method
Step 1.For a given NLPDE with independent variables X = (x, t) and dependent variable u: can be converted to an ODE which transformation ξ = x − ct is wave variable.Also, c is constant to be determined later.
Step 2. We introduce the Riccati equation as following leads to the change of derivatives which admits the use of a finite series of functions of the form where .., m), p, r and q are constants to be determined later.But, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. ( 11).If m is not an integer, then a transformation formula should be used to overcome this difficulty.For aforementioned method, expansion ( 16) reduces to the standard tanh method for Step 3. Substituting Eqs. ( 12) -( 15) into Eq.( 11) with the value of m obtained in Step 2. Collecting the coefficients of Φ k (k = 0, 1, 2, ...), then setting each coefficient to zero, we can get a set of over-determined partial differential equations for a 0 , a i (i = 1, 2, ..., m), b i (i = 1, 2, ..., m) p, q and r with the aid of symbolic computation Maple.
Step 5. We will consider the following twenty seven solutions of generalized Riccati differential equation ( 12) are given in [10,11].
Case 1: For each pq ̸ = 0 or qr ̸ = 0 and ∆ = p 2 − 4qr > 0, Eq. ( 12) has the following solutions where A and B are two nonzero real constants and satisfy Case 2: For each pq ̸ = 0 or qr ̸ = 0 and ∆ = p 2 − 4qr < 0, Eq. ( 12) has the following solutions where A and B are two nonzero real constants and satisfy Case 3: For r = 0 and pq ̸ = 0 Eq. ( 12) has the following solutions where d is an arbitrary constant.Case 4: For r = p = 0 and q ̸ = 0 Eq. ( 12) has the following solution where c is an arbitrary constant.But from (G'/G )-expansion method, we have we set F = G ′ G , then Thus we obtain that the exact solutions derived by (G'/G)-expansion are same as ones by the generalized tanh-coth methods.Hence we use only the generalized (G'/G)-expansion method.

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CONSTRUCTION OF EXACT SOLUTIONS TO THE MODIFIED FORMS OF DP AND CH EQUATIONS

The modified Camassa-Holm equation
We next consider the modified CH equation [40] as The wave variable ξ = x − ct PDE transforms to an ODE where by integrating Eq. ( 66) with respect to ξ and considering the zero constants for integration, we get Applying the procedure given in the previous sections and balancing uu ′′ and u 3 in Eq. ( 67), we obtain m = 2.

Conclusion
The generalized (G'/G)-expansion method was successfully used to establish periodic wave and solitary wave solutions.The obtained results complement the useful works of others for this important equations.Generalized (G'/G)-expansion method is a useful method for finding travelling wave solutions of nonlinear evolution equations.These exact solutions include three types hyperbolic function solution, trigonometric function solution and rational solution.The generalized (G'/G)-expansion method is more powerful in searching for exact solutions of NLPDEs.Some of these results are in agreement with the results reported specially by [40].Also, new results are formally developed in this article.It can be concluded that the this method is a very powerful and efficient technique in finding exact solutions for wide classes of problems.