Journal of Applied Mathematics

Volume 2013 (2013), Article ID 960798, 7 pages

http://dx.doi.org/10.1155/2013/960798

## The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 17 September 2013; Revised 7 November 2013; Accepted 7 November 2013

Academic Editor: Anjan Biswas

Copyright © 2013 Yun-Mei Zhao 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.

#### Abstract

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

#### 1. Introduction

Nonlinear phenomena exist in all areas of science and engineering, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. It is well known that many nonlinear partial differential equations (NLPDEs) are widely used to describe these complex physical phenomena. The exact solution of a differential equation gives information about the construction of complex physical phenomena. Therefore, seeking exact solutions of NLPDEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages, like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [1, 2], the auxiliary equation method [3], the sine-cosine method [4], the Jacobi elliptic function method [5], the exp-function method [6], the tanh-function method [7, 8], the Darboux transformation [9, 10], and the -expansion method [11, 12].

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [13, 14] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [15–19].

In this paper, we first apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained.

#### 2. Description of Methods

##### 2.1. The Simplest Equation Method

*Step 1. *Suppose that we have a nonlinear partial differential equation (PDE) for in the form
where is a polynomial in its arguments.

*Step 2. *By taking , we look for traveling wave solutions of (1) and transform it to the ordinary differential equation (ODE)

*Step 3. *Suppose the solution of (2) can be expressed as a finite series in the form
where satisfies the Bernoulli or Riccati equation, is a positive integer that can be determined by balancing procedure, and are parameters to be determined.

The Bernoulli equation we consider in this paper is
where and are constants. Its solutions can be written as
where , and are constants.

For the Riccati equation
where , and are constants, we will use the solutions
where .

*Step 4. *Substituting (3) into (2) with (4) (or (6)), then the left hand side of (2) is converted into a polynomial in ; equating each coefficient of the polynomial to zero yields a set of algebraic equations for . Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.

*Step 5. *Assuming that the constants can be obtained in Step 4 and substituting the results into (3), then we obtain the exact traveling wave solutions for (1).

##### 2.2. The Modified Simplest Equation Method

In the modified version, one makes an ansatz for the solution as where are arbitrary constants to be determined, such that and is an unspecified function to be determined afterward.

Substitute (8) into (2) and then we account the function . As a result of this substitution, we get a polynomial of and its derivatives. In this polynomial, we equate the coefficients of the same power of to zero, where . This procedure yields a system of equations which can be solved to find , and . Then the substitution of the values of , and into (8) completes the determination of exact solutions of (1).

#### 3. Solutions of the Elliptic-Like Equation

Now, let us choose the following elliptic-like equation where , and are arbitrary constants. Equation (9) is one of the most important auxiliary equations, because many nonlinear evolution equations can be converted to (9) using the travelling wave reduction.

##### 3.1. Using Simplest Equation Method

###### 3.1.1. Solutions of (9) Using the Bernoulli Equation as the Simplest Equation

Considering the homogeneous balance between , and we get , so the solution of (9) is the form

Substituting (10) into (9) and making use of the Bernoulli equation (4) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , and . Solving this system of algebraic equations, with the aid of Maple, we obtain

Therefore, using solutions (5) of (4) and ansatz (10), we obtain the following exact solution of (9):

###### 3.1.2. Solutions of (9) Using Riccati Equation as the Simplest Equation

Suppose the solutions of (9) are the form

Substituting (13) into (9) and making use of the Riccati equation (6) and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , and . Solving this system of algebraic equations, with the aid of Maple, one possible set of values of , and is

Therefore, using solutions (7) of (6) and ansatz (13), we obtain the following exact solution of (9):

##### 3.2. Using Modified Simplest Equation Method

Suppose the solution of (9) is the form where and are constants, such that , and is an unspecified function to be determined. It is simple to calculate that

Substituting the values of , and into (9) and equating the coefficients of , and to zero yield

Solving (18), we obtain

And solving (21), we obtain

*Case 1. *When , we obtain trivial solution; therefore, the case is rejected.

*Case 2. *When , (19) and (20) yield
Integrating (24) with respect to , we obtain
Using (25), from (20), we obtain

Upon integration, we obtain Where and are constants of integration. Therefore, the exact solution of (9) is

From (28), we obtain the exact solution of (9) which is

We can arbitrarily choose the parameters and . Therefore, if we set , (29) reduces to

Again setting , (29) reduces to,

Using hyperbolic function identities, from (30) and (31), we obtain the following periodic solutions

#### 4. Exact Solutions of Some Class of NLPDEs

##### 4.1. The Perturbed Nonlinear Schrödinger's Equation (NLSE) in the Form [20]

Using where is the third order dispersion, is the nonlinear dispersion, while is also a version of nonlinear dispersion. We assume that (33) has exact solution in the form where , and are arbitrary constant to be determined. Substituting (34) into (33), removing the common factor , we have where , and , and are positive constants and the prime means differentiation with respect to . Then we have two equations as follows

Integrating (36) with respect to once and setting the integration constant to be zero, then we have

As (37) and (38) have the same solutions, we have the following equation: where .

From (39), we can obtain

Based on the conclusion just mentioned, we only solve (38) or (37), instead of both (37) and (38), provided that (37) and (36) are replaced by (40), respectively, we get

Equation (41) is identical to (9) and , and are defined by

Then, solutions of (33) are defined as follows: where , appearing in these solutions, is given by relations (12), (15), and (28)–(32). , and are defined by (42).

##### 4.2. The Klein-Gordon-Zakharov (KGZ) System [21]

Consider wherein the complex valued unknown function denotes the fast time scale component of electric field raised by electrons, and the real valued unknown function represents the deviation of ion density. and are some real parameters.

We assume that

Substituting (45) into (44), we have

Integrating (47) with respect to twice and setting the integration constant to be zero, then we have

Substituting (48) into (46), we have

Equation (49) is identical to (9) and , and are defined by

Then, solutions of the Klein-Gordon-Zakharov (KGZ) system are defined as follows: where appearing in these solutions is given by relations (12), (15), and (28)–(32). , and are defined by (50).

##### 4.3. A Class of Nonlinear Partial Differential Equations (NPDEs)

We consider a class of NLPDEs with constant coefficients [22] where are real constants and . Equations (52) are a class of physically important equations. In fact, if one takes then (52) represent the Davey-Stewartson (DS) equations [23]

If one takes then (52) become generalized Zakharov (GZ) equations [24]

Since is a complex function, we assume that where both and are real functions and , and are constants to be determined later. Substituting (57) into (52), we have the following ODE for and :

If we set then (58) reduces to

Integrating (59) twice to , we get where is the integrating constant and Substituting (62) into (61) yields

Equation (63) is identical to (9) and , and are defined by

Then, solutions of (52) are defined as follows: where appearing in these solutions is given by relations (12), (15), and (28)–(32) and , and are defined by (64).

We may obtain from (54) that where is the integrating constant and then (54) reduce to

This equation coincides also with (9), where , and are defined as follows: where , appearing in these solutions is given by relations (12), (15), and (28)–(32) and , and are defined by (68).

We may obtain from (56) that where is the integrating constant and then (56) reduce to

This equation coincides also with (9), where , and are defined as follows: where , appearing in these solutions is given by relations (12), (15), and (28)–(32) and , and are defined by (71).

#### 5. Conclusions

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations, and the elliptic-like equation is one of the most important auxiliary equations because many nonlinear evolution equations, such as the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, the generalized Zakharov equations, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system, and the generalized ZK-BBM equation, can be converted to this equation using the travelling wave reduction.

In this paper, we apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation. The exact solutions of the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, and the generalized Zakharov equations are derived. Comparing the currently proposed method with other methods, such as the -expansion method, the various extended hyperbolic methods, and the exp-function method, we might conclude that some exact solutions that we obtained can be investigated using these methods with the aid of the symbolic computation software, such as Matlab, Mathematica, and Maple to facilitate the complicated algebraic computations. But, by means of the simplest equation method and the modified simplest equation method the exact solutions to these equations have been gained in this paper without using the symbolic computation software since the computations are simple. This study shows that the simplest equation method and the modified simplest equation method are much more simple than the other methods and can be applied to many other nonlinear evolution equations.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193; 2013FZ117), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452; 2013C079).

#### References

- M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,”
*Physics Letters A*, vol. 213, no. 5-6, pp. 279–287, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,”
*Physics Letters A*, vol. 216, no. 1–5, pp. 67–75, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,”
*Physics Letters A*, vol. 309, no. 5-6, pp. 387–396, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Y. Yan and H. Q. Zhang, “New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics,”
*Physics Letters A*, vol. 252, no. 6, pp. 291–296, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Fan and J. Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,”
*Physics Letters A*, vol. 305, no. 6, pp. 383–392, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-H. Wu and J.-H. He, “EXP-function method and its application to nonlinear equations,”
*Chaos, Solitons & Fractals*, vol. 38, no. 3, pp. 903–910, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. A. Abdusalam, “On an improved complex tanh-function method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 99–106, 2005. View at Google Scholar · View at Scopus - A. A. Soliman, “Extended improved tanh-function method for solving the nonlinear physical problems,”
*Acta Applicandae Mathematicae*, vol. 104, no. 3, pp. 367–383, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. B. Leble and N. V. Ustinov, “Darboux transforms, deep reductions and solitons,”
*Journal of Physics A*, vol. 26, no. 19, pp. 5007–5016, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-C. Hu, X.-Y. Tang, S.-Y. Lou, and Q.-P. Liu, “Variable separation solutions obtained from Darboux transformations for the asymmetric Nizhnik-Novikov-Veselov system,”
*Chaos, Solitons & Fractals*, vol. 22, no. 2, pp. 327–334, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. L. Wang, X. Z. Li, and J. L. Zhang, “The $\left(\mathrm{G\text{'}}/G\right)$-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 · View at Google Scholar · View at MathSciNet - S. M. Guo and Y. B. Zhou, “The extended $\left(\mathrm{G\text{'}}/G\right)$-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3214–3221, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - N. A. Kudryashov, “Exact solitary waves of the Fisher equation,”
*Physics Letters A*, vol. 342, no. 1-2, pp. 99–106, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations,”
*Chaos, Solitons and Fractals*, vol. 24, no. 5, pp. 1217–1231, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. K. Vitanov and Z. I. Dimitrova, “Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 10, pp. 2836–2845, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed and S. A. Hoda Ibrahim, “Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method,”
*Chinese Physics Letters*, vol. 29, no. 6, Article ID 060201, 2012. View at Google Scholar - A. J. Mohamad Jawad, M. D. Petković, and A. Biswas, “Modified simple equation method for nonlinear evolution equations,”
*Applied Mathematics and Computation*, vol. 217, no. 2, pp. 869–877, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei, and A. Biswas, “New exact traveling wave solutions for DS-I and DS-II equations,”
*Nonlinear Analysis: Modelling and Control*, vol. 17, no. 3, pp. 369–378, 2012. View at Google Scholar · View at MathSciNet - N. Taghizadeh, M. Mirzazadeh, A. Samiei Paghaleh, and J. Vahidi, “Exact solutions of nonlinear evolution equations by using the modified simple equation method,”
*Ain Shams Engineering Journal*, vol. 3, no. 3, pp. 321–325, 2012. View at Publisher · View at Google Scholar - Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “New exact solutions to the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity,”
*Applied Mathematics and Computation*, vol. 216, no. 10, pp. 3064–3072, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. H. Shi, Q. Xiao, and X. J. Liu, “Extended wave solutions for a nonlinear Klein-Gordon-Zakharov system,”
*Applied Mathematics and Computation*, vol. 218, no. 19, pp. 9922–9929, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,”
*Physics Letters A*, vol. 323, no. 1-2, pp. 77–88, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,”
*Proceedings of the Royal Society A*, vol. 338, pp. 101–110, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Malomed, D. Anderson, M. Lisak, M. L. Quiroga-Teixeiro, and L. Stenflo, “Dynamics of solitary waves in the Zakharov model equations,”
*Physical Review E*, vol. 55, no. 1, pp. 962–968, 1997. View at Publisher · View at Google Scholar · View at Scopus