Explicit and Exact Traveling Wave Solutions of Cahn Allen equation using MSE Method

By using Modified simple equation method, we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. As a result, the existence of solitary wave solutions of the Cahn Allen equation is obtained. Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters. Finally, the variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in mathematical biology, quantum mechanics and plasma physics.


Introduction
The mathematical modeling of events in nature can be explained by differential equations. It is well familiar that various types of the physical phenomena in the field fluid mechanics, quantum mechanics, electricity, plasma physics, chemical kinematics, propagation of shallow water waves and optical fibers are modeled by nonlinear evolution equation and the appearance of solitary wave solutions in nature is somewhat frequent. But, the nonlinear processes are one of the major challenges and not easy to control because the nonlinear characteristic of the system abruptly changes due to some small changes in parameters including time. Thus, the issue becomes more difficult and hence crucial solution is needed. The solutions of these equations have crucial impact in mathematical physics and engineering. The variety of solutions of NLEEs, that are mutual operating different mathematical techniques, is very important in many fields of science such as fluid mechanics, optical fibers, technology of space, control engineering problems, hydrodynamics, meteorology, plasma physics, applied mathematics. Advance nonlinear techniques are major to solve inherent nonlinear problems, particularly those are involving dynamical systems and allied areas. In recent years, there become large improvements in finding the exact solutions of NLEEs. Many powerful methods have been established and enhanced, such as, the modified extended Fan sub-equation method [1], the homogeneous balance method [2][3], the Jacobi elliptic function expansion [4], the Backlund transformation method [5,6], the Darboux transformation method [7],the Adomian decomposition method [8][9], the auxiliary equation method [10,11], the ) / ( G G′ -expansion method [12][13][14][15][16][17][18], the Exp-)) ( [19], the sine-cosine method [20][21][22], the tanh method [23], the F-expansion method [24,25], the exp-function method [26,30], the modified simple equation method [27][28][29], first integral method [31], Simple equation method [32], Bilinear method [33], transformed rational function method [34] and so on. Most of the above methods are depend on computational software except MSE method.
The objective of this paper is to look for new exact traveling wave solutions including topological soliton, single soliton solutions of the well-recognized Cahn-Allen equation [30,31] via MSE method.

Description of the MSE Method
Consider a general form of a nonlinear evolution equation, and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following, we present the main steps of the method: Step 1: Combine the real variables x and t by a compound variable ξ where G is a polynomial in ) (ξ u and its derivatives, wherein and so on.
Step 2: We also consider that the Eq.(3) has the formal solution . ) ( are constants to be determined, and ) (ξ S is also unknown function to be evaluated. Step 3: The value of positive integer n in Eq. (4) can be determined by taking into account the homogeneous balance between the highest order nonlinear terms and the derivatives of highest then the degree of the other expression will be ) (ξ u as follows: Step 4: Inserting Eq. (4) into Eq. (3), we get a polynomial of and its derivatives and ) , , In the resultant polynomial, we equate all the coefficients of to zero. This technique produces a system of algebraic and differential equations which can be solved receiving ), ,

Traveling wave solution of Cahn Allen Equation
Let's consider nonlinear parabolic partial differential equation given by , Eq.(5) becomes Cahn Allen equation [28,29]. This equation arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. To solve this example, we can use transformation wt kx + = ξ (where, k and w are the wave number and the wave speed, respectively) then Eq. (5) becomes to an ordinary differential equation Putting Eq.(6)-Eq.(9) in the Eq. (6) and equating coefficients of like powers of ) ( , we get Coefficient of ( ) : Coefficient of ( ) : Coefficient of ( ) : Coefficient of ( ) : From Eq.(10), we achieve 1 , and from Eq. (13), Integrating we have ), ) 3 ( Using (16) and (17), we attain Here 1 c and 2 c are arbitrary constants.

Case-I: For set
where ) 2 where ) 2 where ) 2 Here 1 c and 2 c are arbitrary constants.
where ) 2 where ) 2 Since  Fig-1 and (24) is similar to the Fig-2 and omitted for convenience.
Again with commercial software we can also find some solutions of the Cahn Allen equation (solving from (11) and (12)). For For

Comparison
In this section, we compare our solution with some well-known methods namely exp-function method and first integral method as follows:

Conclusions
In this article, we have successfully implemented the MSE method to find the exact traveling wave solutions of the Cahn-Allen equation. Comparing the MSE method to other methods, we claim that the MSE method is straightforward, efficient, and can be used in many other nonlinear evolution equations. In the existing methods, such as, the (G'/G)-expansion method, the Expfunction method, the tanh-function method it is required to make use of the symbolic computation software, such as Mathematica or Maple to facilitate the complex algebraic computations. To solve non-linear evolution equations via MSE method no auxiliary equations is needed. On the other hand, via the MSE method, the exact and solitary wave solutions to these equations have been achieved without using any symbolic computation software because the method is very simple and has easy computations.