G. M. Moatimid, M. H. M. Moussa, Rehab M. El-Shiekh, A. A. El-Satar
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo, Egypt
Correspondence to: A. A. El-Satar, Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo, Egypt.
Email: | |
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
Abstract
By using symbolic computation, we apply Auxiliary equation method to construct exact solutions of Non-Linear Klein-Gordon equation. We show that Auxiliary equation method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
Keywords:
Traveling Wave, Exact Solutions And Auxiliary Equation
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
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