Abstract

We study two nonlinear partial differential equations, namely, the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

1. Introduction

The investigation of exact travelling wave solutions of nonlinear partial differential equations (NLPDEs) is important for the understanding of most nonlinear physical phenomena that appear in many areas of scientific fields such as plasma physics, solid state physics, fluid dynamics, optical fibers, mathematical biology, and chemical kinetics [1, 2]. A number of methods have been developed for finding travelling wave solutions to NLPDEs. These include the homogeneous balance method [3], the ansatz method [4, 5], variable separation approach [6], inverse scattering transform method [2], Bäcklund transformation [7], Darboux transformation [8], Hirota bilinear method [9], the -expansion method [10], the reduction mKdV equation method [11], the trifunction method [12, 13], the projective Riccati equation method [14], the sine-cosine method [15, 16], the Jacobi elliptic function expansion method [17, 18], the -expansion method [19], the exp-function expansion method [20], dynamical system method [2123], and Lie symmetry method [2428].

In this paper we study two nonlinear partial differential equations, namely, the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

2. The Symmetric Regularized Long Wave Equation

We first consider that the symmetric regularized long wave equation (SRLW) as given by is a nonlinear evolution equation which arises in several physical applications, for example in sound waves in a plasma [29]. Exact travelling wave solutions of this equation were obtained using the -expansion method [29]. In the present work, Lie symmetry method along with the simplest equation method and the exp-function method are used to construct exact solutions for this equation. First the Lie point symmetries of the SRLW equation (1) are found using the Lie algorithm [25]. These Lie point symmetries are then used to transform (1) into an ordinary differential equation. The simplest equation method [30] and the exp-function method [20] are then used to construct exact solutions of the ordinary differential equation, which leads to the exact solutions of the SRLW equation.

2.1. Lie Point Symmetries of (1) and Symmetry Reduction

The symmetry group of the SRLW equation (1) is generated by the vector field Applying the fourth prolongation of to (1) and solving the resultant overdetermined system of linear partial differential equations, we obtain the following two translation symmetries: Now taking the linear combination of these translation symmetries and , namely, the symmetry , where is a constant, leads to the two invariants Treating as the new dependent variable and as the new independent variable and then substituting the value of into the SRLW equation (1) transform (1) into a fourth-order nonlinear ordinary differential equation:

2.2. Exact Solutions of (1) Using Simplest Equation Method

Now the simplest equation method [30, 31] is used to solve (5), and henceforth one obtains the exact solutions of the SRLW equation (1). The Bernoulli and Riccati equations will be used as the simplest equations. The Bernoulli and Riccati equations are well-known equations whose solutions can be expressed in terms of elementary functions [28].

The Bernoulli equation which we use here is given by where and are constants. Its solution is given by where is a constant of integration [28].

For the Riccati equation where , , and are constants, the solutions to be used are with and being a constant of integration [28].

2.2.1. Solutions of (1) Using Bernoulli as the Simplest Equation

The solutions of the ODE (5) are considered to be in the form where satisfies the Bernoulli or Riccati equations, is a positive integer that can be determined by balancing the highest order derivative term with the highest order nonlinear term [31], and , () are parameters to be determined.

The balancing procedure yields , so the solutions of (5) are of the form Substituting (11) into (5), making use of the Bernoulli equation (6), and then equating all coefficients of the function to zero, we obtain the following algebraic system of equations in terms of , , and : Solving this system, with the aid of Maple, we obtain the following values for the constants: As a result a solution of the symmetric regularized long wave equation (1) using the Bernoulli equation as the simplest equation is where and is a constant of integration.

2.2.2. Solutions of (1) Using Ricatti as the Simplest Equation

The balancing procedure yields , so the solutions of (5) are of the form Substituting (15) into (5), making use of the Ricatti equation (8), and then equating all coefficients of the function to zero, we obtain an algebraic system of equations in terms of , , and . Solving the resultant algebraic equations, we obtain the following set of values: It follows that the solutions for the symmetric regularized long wave equation (1) using the Ricatti equation as the simplest equation are where with and is a constant of integration.

2.3. Solution of (1) Using the Exp-Function Method

In this section we use the exp-function method [20] to solve the symmetric regularized long wave equation (1). We consider solutions of (5) in the form where , , , and are positive integers to be determined and and are arbitrary constants [20]. The balancing procedure of the exp-function method produces and . For simplicity, we set and so that (18) is reduced to Substituting (19) into (5) and solving the resultant ODE, with the help of Maple, one possible set of values of the constants is As a result we obtain the solution

3. The Klein-Gordon-Zakharov Equations

The Klein-Gordon-Zakharov (KGZ) equations [32]

are a coupled system of nonlinear partial differential equations of two functions and . This model describes the interaction of the Langmuir wave and the ion acoustic wave in plasma. The function denotes the fast time scale component of electric field raised by electrons and the function denotes the deviation of ion density from its equilibrium. Here is a complex function and is a real function. Note that if we remove the term , then this system reduces to the classical Klein-Gordon-Zakharov system [33]

A number of studies have been conducted for this system ((23a) and (23b)) in different time space [3438]. However, for the KGZ equations (22a) and (22b), Chen [39] considered orbital stability of solitary waves, while Shi et al. [33] employed the sine-cosine method and the extended tanh method to construct exact solutions of the KGZ equations (22a) and (22b).

In this paper, we employ an entirely different approach, namely, the travelling wave variable approach along with the simplest equation method to obtain exact solutions of the KGZ equations (22a) and (22b).

3.1. Solution of (22a) and (22b) Using the Travelling Wave Variable Approach

The travelling wave variable approach converts the system of nonlinear partial differential equations into a system of nonlinear ordinary differential equations, which we then solve to obtain exact solutions of the system.

In order to solve the KGZ equations (22a) and (22b), we first transform it into a system of nonlinear ordinary differential equations which can then be solved in order to obtain its exact solutions.

We make the wave variable transformation where , , , and are real constants and . Using this transformation, (22a) and (22b) transform into

Integrating (25b) twice and taking the constants of integration to be zero, we obtain Now substituting (26) into (25a), we get which can be written in the form where Solving (28), with the aid of Mathematica, we obtain the solution where is a Jacobian elliptic function of the sine amplitude [40], is the modulus of the elliptic function with . Here and are constants of integration. Reverting back to our original variables, we can now write the solution of our Klein-Gordon-Zakharov equations as where and and are as above.

Now can be obtained from (26).

It should be noted that the solution (32) is valid for , as approaches zero, the solution becomes the normal sine function, , and as approaches 1, the solution tends to the function, .

The profile of the solution (32) is given in Figure 1.

3.2. Solutions of (22a) and (22b) Using the Simplest Equation Method

We consider the solutions of (27) in the form where satisfies the Bernoulli or the Riccati equation.

3.2.1. Solutions of (22a) and (22b) Using Bernoulli as the Simplest Equation

We consider the Bernoulli equation where and are constants.

The balancing procedure yields , so the solution of (27) is of the form Substituting (36) into (27), making use of the Bernoulli equation (35), and then equating all coefficients of the function to zero, we obtain the following algebraic system of equations: Solving this system, with the aid of Maple, we obtain the following values for the constants: As a result, a solution of the Klein-Gordon-Zakharov equations (22a) and (22b), using the Bernoulli equation as the simplest equation, is where is a constant of integration.

3.2.2. Solutions of (22a) and (22b) Using Riccati as the Simplest Equation

We use the Riccati equation given by where , , and are constants. The balancing procedure yields , so the solution of (27) is of the form Similar calculations yield the following set of values: As a result the two solutions of (22a) and (22b) are where and . is given by , is a constant of integration, and and are as obtained above.

It should be noted that by substituting the above value of into (26), one can now obtain the solution for the variable .

4. Conclusion

In this paper we studied two nonlinear partial differential equations. Firstly, Lie symmetry approach along with the simplest equation and the Exp-function method were used to obtain travelling wave solutions of the symmetric regularized long wave equation. Secondly, the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

Conflict of Interests

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

Acknowledgments

Isaiah Elvis Mhlanga would like to thank the Faculty Research Committee, FAST, North-West University, Mafikeng Campus, for the financial support. Chaudry Masood Khalique thanks the North-West University, Mafikeng Campus, for its continued support.