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 equationmethod and themodified simplest equationmethod, 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.


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 tanhfunction 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][16][17][18][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.

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.

Journal of Applied Mathematics
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 ( = 0, 1, 2, . . . , ) 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 1 , 2 , and 0 are constants. For the Riccati equation where , , and are constants, we will use the solutions where 2 = 2 − 4 .

The Modified Simplest Equation Method
. In the modified version, one makes an ansatz for the solution ( ) as where ( = 0, 1, 2, . . . , ) are arbitrary constants to be determined, such that ̸ = 0 and ( ) is an unspecified function to be determined afterward.

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.

Solutions of (9) Using the Bernoulli Equation as the Simplest Equation.
Considering the homogeneous balance between ( ), and 3 ( ) we get = 1, 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 ( = 0, 1), , 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):

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 ( = 0, 1), , , and . Solving this system of algebraic equations, with the aid of Maple, one possible set of values of ( = 0, 1), , , and is Therefore, using solutions (7) of (6) and ansatz (13), we obtain the following exact solution of (9):

Using Modified Simplest Equation Method
. Suppose the solution of (9) is the form where 0 and 1 are constants, such that 1 ̸ = 0, and ( ) is an unspecified function to be determined. It is simple to calculate that Substituting the values of , , and 3 into (9) and equating the coefficients of 0 , −1 , −2 , and −3 to zero yield Solving (18), we obtain And solving (21), we obtain Case 1. When 0 = 0, we obtain trivial solution; therefore, the case is rejected.

The Perturbed Nonlinear Schrödinger's Equation (NLSE) in the Form
where 1 is the third order dispersion, 2 is the nonlinear dispersion, while 3 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 exp( ( − )), we have where ( = 1, 2, and 3), , 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: From (39), we can obtain Journal of Applied Mathematics 5 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). [21]. Consider

The Klein-Gordon-Zakharov (KGZ) System
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).

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.