1. Introduction

In recent years, the exact solutions of nonlinear PDEs have been investigated by many authors ( see for example [1-30]^[1] ) who are interested in nonlinear physical phenomena. Manäcklund transformation ^[2], Darboux transformation ^[3], Hirota bilinear method ^[4], variable separation approach ^[5], various tanh methods ^{[6, 7, 8, 9]}, homogeneous balance method ^[10] , similarity reductions method ^{[11, 12]} , the reduction mKdV equation method ^[13], the tri-function method ^{[14, 15]}, the projective Riccati equation method ^[16], the Weierstrass elliptic function method ^[17], the Sine- Cosine method ^{[18, 19]}, the Jacobi elliptic function expansion ^{[20, 21]}, the complex hyperbolic function method ^[22], the truncated Painleve’ expansion ^[23], the F-expansion method ^[24], the rank analysis method ^[25] and so on.

In the present paper, we shall use a simple method which is called the -expansion method ^{[26, 27]}. This method is firstly proposed by the Chinese Mathematicians Wang et al ^[28] for which the traveling wave solutions of nonlinear equations are obtained. The main idea of this method is that the traveling wave solutions of nonlinear equations can be expressed by a polynomial in , where satisfies the second order linear ordinary differential equation where ,where and c are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in the given nonlinear equations .The coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the proposed method. This new method will play an important role in expressing the traveling wave solutions for nonlinear evolution equations via the NLS equation and the RLW equation in terms of hyperbolic, trigonometric and rational functions.

2. Description of the (G’/G)-Expansion Method

Suppose we have the following nonlinear PDE:

(1)

where is an unknown function, P is a polynomial in and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation method :

Setp1. The traveling wave variable

(2)

where k and c are the wave number and the wave speed, respectively. Permits us reducing Equation.(1) to an ODE for in the form

(3)

where ,

Setp2. Suppose that the solution Equation.(3) can be expressed by a polynomial in as follows:

(4)

while satisfies the second order linear differential equation in the form:

(5)

where and are constants to be determined later.

Setp3. The positive integer “n” can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in Equation.(3). Therefore, we can get the value of n in Equation.( 4).

Setp4. Substituting Equation.( 4) into Equation.( 3) and using Equation.(5), collecting all terms with the same power of together and then equating each coefficient of the resulted polynomial to zero, yield a set of algebraic equations for and k.

Setp5. Solving the algebraic equations by use of Maple or Mathematica, we obtain values for and k.

Setp6. Since the general solutions of Equation. (5) have been well known for us, then substituting the obtained coefficients and the general solution of Equation.(5) into Equation. (4), we have the travelling wave solutions of the nonlinear PDE (1).

3. Applications of the Method

In this section, we apply the -expansion method to construct the traveling wave solutions for some nonlinear PDEs, namely the equation and the RLW equation which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers.

We start with the following equation ^[29] in the form:

(6)

where p is a positive constant.

Now we convert Equation.(6) into the real form :

(7)

The traveling wave variable (2) permits us converting Equation.(7) into the following ODEs.

(8)

Suppose that the solution of ODEs(8) can be expressed by polynomial in terms of as follows:

(9)

where and are arbitrary constants, while satisfies the second order linear ODE (5).Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Equation.(9), we get . Thus, we have

(10)

where and are constants to be determined later. Substituting Equation.(10) with Equation. (5) into Equation.(8) and collecting all terms with the same power of . Setting each coefficients of this polynomial to be zero, we have the following system of algebraic equations:

(11)

On solving the above algebraic Equations.(11) by using the Maple or Mathematica, we have:

(12)

where and .

Substituting Equation.(12) into Equation.(10) yields

(13)

where

(14)

On solving Equation.( 5), we deduce that

(15)

where A and B are arbitrary constants and .

On substituting Equation.(15) into Equation.(13), we deduce the following two types of traveling wave solutions:

Case 1.  then we have the hyperbolic solution

(16)

In this case the hyperbolic solutions of Equation.(6) take the following form:

(17)

Case 2.  then we have the trigonometric solution.

(18)

In this case the hyperbolic solutions of Equation.(6) take the following form:

(19)

In particular, if we set in Equation.(16), then we get

(20)

The traveling wave solution of Equation.(6) take the following form:

(21)

If then we deduce that:

(22)

The traveling wave solution of Equation.(6) take the following form:

(23)

where .The solutions (21) and (23) represent the solitary wave solutions of Equation.(6).

In this subsection, we study the RLW equation ^[30] in the form :

(24)

where and are real constants. The regularized-long-wave (RLW) equation was first obtained by Peregrine [3l]^[31] to describe the development of an undular bore; i.e, a smooth solitary wave that is observed to propagate in shallow water channels. Since then, the RLW equation has been used as a one-dimensional model for drift waves in plasmas . It is also appropriate to describe Rossby waves in geophysics. Let us now solve Equation.(24) by the proposed method. To this end, we see that the traveling wave variable (2) permits us converting Equation.(24) into the following ODE:

(25)

integrating Equation.(25) with respect to once, yields

(26)

where is an integration constant.

Considering the homogeneous balance between highest order derivatives and nonlinear terms in Equation.(26), we get .Thus,we have

(27)

where and are arbitrary constants to be determined later. Substituting Equation.(27) with Equation.(5) into Equation.(26) , collecting all terms with the same power of and setting them to zero, we have the following system of algebraic equations:

(28)

Solving the above algebraic equations (28) by using the Maple or Mathematica, yields.

(29)

Substituting Equation.(29) into Equation.(27) we obtain

(30)

where

(31)

From Equation.(15) and Equation.(30), we deduce the following three types of traveling wave solutions:

Case 1.  then we have the hyperbolic solution

(32)

Case 2.  then we have the trigonometric solution

(33)

Case 3.  then we have the rational solution

(34)

In particular if and then we deduce from Equation.(32) that:

(35)

while, if then we deduce that:

(36)

where . The solutions (35) and (38) represent the solitary wave solutions of Equation.(24).

4. Conclusion

In this paper, we have seen that three types of traveling wave solutions in terms of hyperbolic, trigonometric and rational functions for the equation and the RLW equation are successfully found out by using the -expansion method. From our results obtained in this paper, we conclude that the -expansion method is powerful, effective and convenient. The performance of this method is reliable, simple and gives many new solutions. The -expansion method has more advantages: It is direct and concise. Also, the solutions of the proposed nonlinear evolution equations in this paper have many potential applications in physics and engineering. Finally, this method provides a powerful mathematical tool to obtain more general exact solutions of a great many nonlinear PDEs in mathematical physics.

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