Research Article | Open Access

# Soliton Solutions of the Klein-Gordon-Zakharov Equation with Power Law Nonlinearity

**Academic Editor:**V. Rai

#### Abstract

We introduce a new version of the trial equation method for solving nonintegrable partial differential equations in mathematical physics. Some exact solutions including soliton solutions and rational and elliptic function solutions to the Klein-Gordon-Zakharov equation with power law nonlinearity in (1 + 2) dimensions are obtained by this method.

#### 1. Introduction

In recent years there have been many works on the qualitative research of the global solutions for the Klein-Gordon-Zakharov (KGZ) equations [1–4]. Chen considered orbital stability of solitary waves for the KGZ equations in [5]. More recently, some exact solutions for the Zakharov equations are obtained by using different methods [6–9]. These solutions are not general and by no means exhaust all possibilities. They are only some particular solutions within some specific parameters choices.

The aim of this paper is to find the new and more general explicit and exact special solutions of the KGZ equations. We obtain various of explicit and exact special solutions of the KGZ equations by using the extended trial equation method. These solutions include that of the solitary wave solutions of the singular traveling wave solutions and solitary wave solutions of rational function type.

Solving nonlinear evolution equations has become a valuable task in many scientific areas including applied mathematics as well as the physical sciences and engineering. Many powerful methods, such as the Backlund transformation, the inverse scattering method [10], bilinear transformation, the tanh-sech method [11], the extended tanh method, the pseudospectral method [12], the trial function and the sine-cosine method [13], Hirota method [14], tanh-coth method [15, 16], the exponential function method [17], -expansion method [18, 19], homogeneous balance method [20], and the trial equation method [21–30] have been used to investigate nonlinear partial differential equations problems. There are a lot of nonlinear evolution equations that are integrated using these and other mathematical methods.

In this paper, KGZ equations will be studied by extended trial equation. By virtue of the solitary wave ansatz method, an exact soliton solution will be obtained. The extended trial equation method will be employed to back up our analysis in obtaining exact solutions with distinct physical structures.

#### 2. The Extended Trial Equation Method

The main steps of an extended trial equation method for the nonlinear partial differential equations with higher order nonlinearity are outlined as follows.

*Step 1. *For a given nonlinear partial differential equation with rank inhomogeneous
take the wave transformation
where and . Substituting (2) into (1) yields a nonlinear ordinary differential equation

*Step 2. *Take transformation and trial equation as follows:
in which
where , , and are constants. Using the relations (4) and (5), we can find
where and are polynomials. Substituting these terms into (3) yields an equation of polynomial of as follows:
According to the balance principle we can determine a relation of , , and . We can take some values of , , and .

*Step 3. *Let the coefficients of all be zero, this will yield the following algebraic equations system:
Solving this equation system (8), we will determine the values of ; , and .

*Step 4. *Reduce (5) to the elementary integral form as follows:
Using a complete discrimination system for polynomial to classify the roots of , we solve the infinite integral (9) and obtain the exact solutions to (3). Furthermore, we can write the exact traveling wave solutions to (1), respectively.

#### 3. Mathematical Analysis

We introduce the KGZ equation with power law nonlinearity in dimensions and its soliton solution by extended trial equation method and show its numerical solution at a fixed point.

##### 3.1. The KGZ Equation in (1 + 2) Dimensions

The dimensionless form of the KGZ equation in dimensions that will be studied in this subsection is given by [31] Here, the dependent variables are and , while the independent variables are , , and which are, respectively, referred to as the spatial variables and temporal variable. Power law nonlinearity arises in nonlinear plasmas that solves the problem of small K-condensation in weak turbulence theory. It also arises in the context of nonlinear optics. The parameter dictates the power law nonlinearity, while and are constants. Here, in (10) and (11), is a complex valued function while is a real valued function. Equations (10) and (11) together appear in the area of Plasma Physics. They describe the interaction of Langmuir waves and ion-acoustic waves in plasmas [32, 33]. For solving (10) and (11) with the trial equation method, using the wave variables where , , , , , , and are real constants, (10) and (11) are converted to the system of ODEs where primes denote the derivatives with respect to . Equation (15) is then integrated term by term two times where integration constants are considered zero. This converts it into Substituting (16) into (14) gives Equation (17), with the transformation reduces to where Substituting (6) into (19) and using balance principle yields . If we take , , and , then where and . Solving the algebraic equation system (8) yields Also from (13), it can be seen that . Substituting these results into (5) and (9), we can write where

Integrating (23), we obtain the solutions to (10) and (11) as follows: where Also , , , and are the roots of the polynomial equation Substituting the solutions (25)–(28) into (4) and (18), we obtain, respectively, If we take and , then the solutions (32)–(41) can reduce to rational function solutions traveling wave solutions and soliton solutions where Here, and are the amplitudes of the solitons, and are the inverse widths of the solitons in the - and -directions, respectively, while is the velocity of the soliton. Also, and are the soliton frequencies in the - and -directions, respectively, and is the soliton wave number, while is the phase constant. Thus, we can say that the solitons exist for .

*Remark.* Ismail and Biswas obtained 1-soliton solution of KGZ equation in [31], we obtained soliton solution of this equation. In case , our soliton solutions in (44) and (45) reduce to 1-soliton solutions in [31].

In Figures 1, 2, and 3, we give profiles of numerical soliton solutions of (44) and (45) for various values of parameters.

#### 4. Conclusion

We adopt the extended trial equation method to obtain soliton solutions of the KGZ equations in plasma physics. We obtain some more general solitary wave solutions of the KGZ equations. It not only produces the same solutions but also can pick up what we believe to be new solutions missed by other authors. The results indicate the KGZ equations admit soliton solutions with some arbitrary parameters. The type of exact solitary wave solution is different along with different value of arbitrary parameters. So we can choose appropriate parameter value to obtain solutions which we need in applications. The method can also be employed to solve a large number of other nonlinear evolution equations, such as nonlinear reaction-diffusion equation, the long-short wave resonance equation, the shallow water wave equation, Whitham-Broer-Kaup equation, variant Boussinesq equation, double Sine-Gardon equation, and Dodd-Bullough-Mikhailov equation.

#### References

- B. L. Guo and G. W. Yuan, “Global smooth solution for the Klein-Gordon-Zakharov equations,”
*Journal of Mathematical Physics*, vol. 36, no. 8, pp. 4119–4124, 1995. View at: Google Scholar - T. Ozawa, K. Tsutaya, and Y. Tsutsumi, “Normal form and global solutions for the Klein-Gordon-Zakharov equations,”
*Annales de l'Institut Henri Poincaré Analyse Non Linéaire*, vol. 12, no. 4, pp. 459–503, 1995. View at: Google Scholar - K. Tsutaya, “Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 27, no. 12, pp. 1373–1380, 1996. View at: Publisher Site | Google Scholar - G. Adomian, “Non-perturbative solution of the Klein-Gordon-Zakharov equation,”
*Applied Mathematics and Computation*, vol. 81, no. 1, pp. 89–92, 1997. View at: Google Scholar - L. Chen, “Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations,”
*Acta Mathematicae Applicatae Sinica*, vol. 15, no. 1, pp. 54–64, 1999. View at: Google Scholar - J. L. Zhang, M. L. Wang, D. M. Chen, and Z. D. Fang, “The periodic wave solutions for two nonlinear evolution equations,”
*Communications in Theoretical Physics*, vol. 40, no. 2, pp. 129–132, 2003. View at: Google Scholar - D. J. Huang and H. Q. Zhang, “Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations,”
*Acta Physica Sinica*, vol. 53, no. 8, pp. 2434–2438, 2004. View at: Google Scholar - Y. Chen and B. Li, “New exact travelling wave solutions for generalized Zakharov-Kuzentsov equations using general projective riccati equation method,”
*Communications in Theoretical Physics*, vol. 41, no. 1, pp. 1–6, 2004. View at: Google Scholar - Y. Chen, B. Li, and H.-Q. Zhang, “Bäcklund transformation and exact solutions for a new generalized Zakharov-Kuzentsov equation with nonlinear terms of any order,”
*Communications in Theoretical Physics*, vol. 39, no. 2, pp. 135–140, 2003. View at: Google Scholar - M. J. Ablowitz, P. A. Clarkson, and Solitons,
*Nonlinear Evolution Equations and Inverse Scattering*, Cambridge University press, Cambridge, UK, 1991. - A.-M. Wazwaz, “The tanh method for traveling wave solutions of nonlinear equations,”
*Applied Mathematics and Computation*, vol. 154, no. 3, pp. 713–723, 2004. View at: Publisher Site | Google Scholar - P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,”
*Physical Review Letters*, vol. 70, no. 5, pp. 564–567, 1993. View at: Publisher Site | Google Scholar - A.-M. Wazwaz, “An analytic study of compactons structures in a class of nonlinear dispersive equations,”
*Mathematics and Computers in Simulation*, vol. 63, no. 1, pp. 35–44, 2003. View at: Publisher Site | Google Scholar - R. Hirota, “Exact solution of the korteweg-de vries equation for multiple Collisions of solitons,”
*Physical Review Letters*, vol. 27, no. 18, pp. 1192–1194, 1971. View at: Publisher Site | Google Scholar - W. Malfliet and W. Hereman, “The tanh method: I. Exact solutions of nonlinear evolution and wave equations,”
*Physica Scripta*, vol. 54, no. 6, pp. 563–568, 1996. View at: Google Scholar - M. A. Abdou, “The extended tanh method and its applications for solving nonlinear physical models,”
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 988–996, 2007. View at: Publisher Site | Google Scholar - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons and Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site | Google Scholar - M. Wang, X. Li, and J. Zhang, “The (
*G′/ G*)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,”*Physics Letters A*, vol. 372, no. 4, pp. 417–423, 2008. View at: Publisher Site | Google Scholar - G. Ebadi and A. Biswas, “The (
*G′/ G*) method and topological soliton solution of the K(m,n) equation,”*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 6, pp. 2377–2382, 2011. View at: Google Scholar - M. Wang, “Solitary wave solutions for variant Boussinesq equations,”
*Physics Letters A*, vol. 199, no. 3-4, pp. 169–172, 1995. View at: Google Scholar - C.-S. Liu, “Trial equation method and its applications to nonlinear evolution equations,”
*Acta Physica Sinica*, vol. 54, no. 6, pp. 2505–2509, 2005. View at: Google Scholar - C.-S. Liu, “A new trial equation method and its applications,”
*Communications in Theoretical Physics*, vol. 45, no. 3, pp. 395–397, 2006. View at: Publisher Site | Google Scholar - C.-S. Liu, “Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications,”
*Communications in Theoretical Physics*, vol. 45, no. 2, pp. 219–223, 2006. View at: Publisher Site | Google Scholar - C. S. Liu, “Using trial equation method to solve the exact solutions for two kinds of KdV equations with variable coefficients,”
*Acta Physica Sinica*, vol. 54, no. 10, pp. 4506–4510, 2005. View at: Google Scholar - C.-S. Liu, “Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations,”
*Computer Physics Communications*, vol. 181, no. 2, pp. 317–324, 2010. View at: Publisher Site | Google Scholar - Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics,”
*Pramana*, vol. 77, no. 6, pp. 1023–1029, 2011. View at: Publisher Site | Google Scholar - Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of an irrational trial equation method to high-dimensional nonlinear evolution equations,”
*Journal of Advanced Mathematical Studies*, vol. 5, no. 1, pp. 41–47, 2012. View at: Google Scholar - Y. Pandir, Y. Gurefe, U. Kadak, and E. Misirli, “Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 478531, 16 pages, 2012. View at: Publisher Site | Google Scholar - Y. Gurefe, E. Misirli, A. Sonmezoglu, and M. Ekici, “Extended trial equation method to generalized nonlinear partial differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 10, pp. 5253–5260, 2013. View at: Google Scholar - Y. Gurefe, E. Misirli, Y. Pandir, A. Sonmezoglu, and M. Ekici, “New exact solutions of the davey-stewartson equation with power-law nonlinearity,”
*Bulletin of the Malaysian Mathematical Sciences Society*. In press. View at: Google Scholar - M. S. Ismail and A. Biswas, “1-Soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity,”
*Applied Mathematics and Computation*, vol. 217, no. 8, pp. 4186–4196, 2010. View at: Publisher Site | Google Scholar - J. Wang, “Solitary wave propagation and interactions for the Klein-Gordon-Zakharov equations in plasma physics,”
*Journal of Physics A*, vol. 42, no. 8, pp. 085205–085221, 2009. View at: Google Scholar - T. Wang, J. Chen, and L. Zhang, “Conservative difference methods for the Klein-Gordon-Zakharov equations,”
*Journal of Computational and Applied Mathematics*, vol. 205, no. 1, pp. 430–452, 2007. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2013 Mehmet Ekici et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.