1. Introduction

The investigation of the travelling wave solutions for non-linear partial differential equations plays an important role in the study of non-linear physical phenomena. Non-linear wave phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Non-linear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in non-linear wave equations. In recent years, new exact solutions may help to find new phenomena. A variety of powerful methods, such as inverse scattering method[1, 2], bilinear transformation[3], the tanh-sech method[4-6], extended tanh method[7,8], sine-cosine method[9,10], homogeneous balance method[11], Exp-function method[12,13], improved tanh-function method[14] and Auxiliary equation method[15] were used to develop non-linear dispersive and dissipative problems.

2. Auxiliary Equation Method

Consider a given nonlinear wave equation

(2.1)

we seek its wave solutions.

(2.2)

Consequently, (1.1) is reduced to the ordinary differential equation (ODE):

Auxiliary equation method is based on the assumption that the travelling wave solutions can be expressed in the following form

(2.4)

where φ(η) satisfies Auxiliary equation method

(2.5)

3. Non-Linear Klein-Gordon Equation

We study the following well-known the nonlinear Klein-Gordon equation:

(3.1)

for .

To Higgs equation and for c = 0, it assumes the form of Yang-Milles equation. The Klein- Gordon equation has been studied in many literatures, e.g.,[16-18]. However, numerical treatment for Klein--Gordon equation is rarely reported (cf.[19-21]). Particularly,[20] concern with the decomposition method and difference method used in[21] nonlinear problems. We take u=U+i V, then by used he transformation (2.2) we get:

(3.2)

Now balancing U′′ with U³and V′′ with U³V gives M=1, N=1. Therefore we may choose:

(3.3)

Substituting from equations (3.3) in equation (3.1), collect the coefficients of φ(η)ⁱ (i = 0, ..., 3) and equated them to zero we obtain the system

(3.4)

Solving the system of algebraic equations with the aid of Maple, in Eq.(3.3), we obtain the following results:

(3.5)

Substituting these results into (3.3) and with the aid of Appendix A, we obtain the following multiple soliton-like and triangular periodic solutions for (DSW) system:

(3.6)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

Some trigonometric-function solutions of Eq. (3.1) can be obtained in the limited case when the modulus m → 0(see Appendix B). For example,

4. Conclusions

In this study, we have applied Auxiliary equation method to obtain the generalized solitary wave solutions of Non-Linear Klein-Gordon equation. As we can see in the example of (K. G.) equation, the main advantage of this method over the other methods is that it can be applied to a wide class of nonlinear evolution equations including those in which the odd and even-order derivative terms are coexist. It may be concluded that, Auxiliary equation method can be easily extended to all kinds of nonlinear equations.

Appendix A