Research Article | Open Access
The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations
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.
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 , the sine-cosine method , the Jacobi elliptic function method , the exp-function method , 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.
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
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
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.
Upon integration, we obtain Where and are constants of integration. Therefore, the exact solution of (9) is
We can arbitrarily choose the parameters and . Therefore, if we set , (29) reduces to
Again setting , (29) reduces to,
4. Exact Solutions of Some Class of NLPDEs
4.1. The Perturbed Nonlinear Schrödinger's Equation (NLSE) in the Form 
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
From (39), we can obtain
4.2. The Klein-Gordon-Zakharov (KGZ) System 
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
Integrating (47) with respect to twice and setting the integration constant to be zero, then we have
4.3. A Class of Nonlinear Partial Differential Equations (NPDEs)
We consider a class of NLPDEs with constant coefficients  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 
If we set then (58) reduces to
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.
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).
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