Extended Jacobian Elliptic Function Expansion Method and Its Applications in Mathematical Physics

No one can deny the important role which played by the nonlinear partial differential equations in the description of many and a wide variety of phenomena not only in physical phenomena, but also in plasma, fluid mechanics, optical fibers, solid state physics, chemical kinetics and geochemistry phenomena. So that, during the past five decades, a lot of method was discovered by a diverse group of scientists to solve the nonlinear partial differential equations. Such methods are tanh sech method [1]-[3], extended tanh method [4]-[6], sine cosine method [7]-[9], homogeneous balance method [10, 11], F-expansion method [12]-[14], exp-function method [15, 16], trigonometric function series method [17], G′ G -expansion method [18]-[21], the exp (-φ (ξ))-expansion method [22]-[24],extended Jacobi elliptic function method [25]-[28], the exp (φ (ξ))-expansion method [26]-[28] and so on. The objective of this article is to apply an extended Jacobian elliptic function expansion for finding the exact traveling wave solution of the nonlinear PhiFour equation which play an important role in mathematical physics.

The rest of this paper is organized as follows: In Section 2, we give the description of an extended Jacobian elliptic function expansion.In Section 3, we use this method to find the exact solutions of the nonlinear evolution equations pointed out above.In Section 5, we give the physical interpretations of the solutions.In Section 5, conclusions are given.

2-Description of method
Consider the following nonlinear evolution equation P u, u t , u x , u tt , u xx , … . .= 0, (2.1) since, P is a polynomial in u(x, t) and its partial derivatives.In the following, we give the main steps of this method Step 1.We use the traveling wave solution in the form where c is a positive constant, to reduce Eq.(2.1) to the following ODE: where P is a polynomial in u (ξ) and its total derivatives, while  ′ =   .
Step 2.Making good use of ten Jacobian elliptic functions we assume that (2.3) have the solutions in these forms: Where, , are the Jacobian elliptic sine function, thejacobian elliptic cosine function and the Jacobian elliptic function of the third kind and other Jacobianfunctions which are denoted by Glaisher's symbols and are generated by these three kinds of functions, namely: those have the relations with the modulus m (0 < m < 1).In addition we know that The derivatives of other Jacobian elliptic functions are obtained by using Eq.(1.8).Tobalance the highest order linear term with nonlinear term we define the degree of u asD[u] = n which gives rise to the degrees of other expressions as
According the rules, we can balance the highest order linear term and nonlinear term in Eq. (2.3) so that n in Eq.(2.4) can be determined.In addition we see that when  ⟶ 1 , , ,  degenerate as ℎ, ℎ, ℎ respectively, while when therefore Eq.(2.5) degenerate as the following forms Therefore the extended Jacobian elliptic function expansion method is more general than sine-cosine method, the tanfunction method and Jacobian elliptic function expansion method.

Application
Here, we will apply, an extended Jacobian elliptic function expansion method described in Sec.2 to find the exact traveling wave solutions and the solitary wave solutions of the nonlinear Phi-Four equation [32,33].We consider the nonlinear Phi-Four equation.2) and collecting all term with the same power of  3 ,  2 ,  2 , , , ,  0 we get: Solving above system by using maple 16, we get: Thus the exact traveling wave solution is When  ⟶ 1 we get: Case2: Thus the exact traveling wave solution is

3-Conclusion
An extended Jacobian elliptic function expansion method has been applied in this paper to find the exact traveling wave solutions and then the solitary wave solutions of the nonlinear Phi-Four equation.Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: Our results of the nonlinear Phi-Four equationare new and different from those obtained in [32,33], and figures show the solitary traveling wave solution ofthe nonlinear Phi-Four equation.We can conclude that an extended Jacobian elliptic 4) With I S S N 2347-3487 V o l u m e 1 1 , N u m b e r 8 J o u r n a l o f A d v a n c e s i n P h y s i c s −   −  +  3 = 0, (3.1) whereα and β are real constants.By using the wave transformation u ξ = u x, t , since ξ = x − t ,we get: 1 −   ′′ −  +  3 = 0. (3.2) Balancing between the highest order derivatives and nonlinear terms appearing in Eq. (3.2)  3  ′′ ⇒  = 1 .So that, by using Eq.(2.4) we get the formal solution of Eq. (3.2)   =  0 +  1  +  1 .(3.3) Substituting Eq. (3.5) and its derivative into Eq.(3.

I
S S N 2347-3487 V o l u m e 1 1 , N u m b e r 8 J o u r n a l o f A d v a n c e s i n P h y s i c s ,