Analytical solutions of Phi-four equation

Nonlinear evolution equations (NLEEs) are considerably used to identify a variety of physical circumstances in the areas such as quantum field theory, hydrodynamics, chemical kinematics, geochemistry, electricity, elastic media and plasma physics. Recently, many researchers have introduced a lot of methods to acquire exact solutions of NLEEs such as G'/G-expansion method [1], modified extended tanhfunction method [2], sine-cosine method [3], expfunction method [4], modified simple equation method [4], extended trial equation method [5], generalized Kudryashov method [6] . In this study, MEFM [7] will be implemented to find new analytical solutions of Phi-four equation. We consider Phi-four equation [8-11], 3 0, 0, tt xx u au u u a      (1)


Introduction
Nonlinear evolution equations (NLEEs) are considerably used to identify a variety of physical circumstances in the areas such as quantum field theory, hydrodynamics, chemical kinematics, geochemistry, electricity, elastic media and plasma physics.Recently, many researchers have introduced a lot of methods to acquire exact solutions of NLEEs such as G'/G-expansion method [1], modified extended tanhfunction method [2], sine-cosine method [3], expfunction method [4], modified simple equation method [4], extended trial equation method [5], generalized Kudryashov method [6] .In this study, MEFM [7] will be implemented to find new analytical solutions of Phi-four equation.We consider Phi-four equation [8][9][10][11], 3 0, 0, where a is real constant.This equation can be investigated as a special form of the Klein-Gordon equation that patterns the phenomenon in particle physics where kink and anti-kink solitary waves interact [12].Many scientists have used exact and numerical solutions of Phi-four equation to research some methods such as the sine-cosine method [8], the auxiliary equation method [9], the modified simple equation method [10], homotopy perturbation method [11], homotopy analysis method [11] and Adomian decomposition method [11].In this article, the basic interest is to construct new exact solutions of Phi-four equation via MEFM.In Sec. 2, we clarify basic facts of MEFM.In Sec. 3, we find new exact solutions of the Phi-four equation via MEFM.

Basic facts of method
The fundamental properties of MEFM are introduced in this section.MEFM is predicated on the exp -expansion function method [13][14][15][16].In order to implement this method to the nonlinear partial differential equations, we handle it as follows:   , , , , , 0, where   , u u x t  is an unknown function, P is a polynomial in   , u x t and its derivatives, in which the highest order derivatives and nonlinear terms are included and the subscripts demonstrate the partial derivatives.The fundamental stages of the method are defined as follows: Step 1: Let us investigate the following traveling transformation identified by Using Eq. (3), we can turn Eq. ( 2) into a nonlinear ordinary differential equation (NODE) described by:   , , , , 0, where NODE is a polynomial of U and its derivatives and the superscripts demonstrate the ordinary derivatives according to  .
Step 2: Assume the traveling wave solution of Eq. ( 4) can be shown as follows: where   , , 0 ,0 are constants to be described later, such that 0, 0, There are the following solution families of Eq. ( 6): B E  are constants to be described later.The positive integers N and M can be identified by taking into consideration the homogeneous balance between the highest order derivatives and the nonlinear terms arising in Eq. ( 5).

 
We compensate all the coefficients of same power of to zero.This process provides a system of equations which can be solved to obtain , , , , , , , , , , B E  into Eq. ( 5) the general solutions of Eq. ( 5) fulfil the determination of the solution of Eq. (1).

MEFM for Phi-Four Equation
In this section, we look for the exact solutions of Eq.
(1) by using MEFM.We find the travelling wave solutions of Eq. ( 1) by using the wave variables where k and c are arbitrary constants.
Using balance principle in Eq. ( 14), we obtain where 2 0 A  and 1 0 B  .When we use Eq.( 16) and Eq.(18) in Eq.( 14) we get a system of algebraic equations from the coefficients of polynomial of     exp .

 
By solving this system of algebraic equations by using Wolfram Mathematica 9, it yields us the following coefficients: Case 2: Case 3: ,, 2 ,.

Conclusion
In this paper, we use MEFM to find exact solutions of Phi-four equation.Then, in Figures 1-4, we plot 2D and 3D surfaces of dark soliton solutions and trigonometric function solution of Phi-four equation by using Mathematica Release 9.According to these data and observation, it has been deduced that this method has been influential for the exact solutions of these NLEEs and this method is highly effective and dependable in the sense that reaching analytical solutions.Thus, we can say that this method has a substantial position to attain exact solutions of NLEEs.