Exact Solutions of the Symmetric Regularized Long Wave Equation and the Klein-Gordon-Zakharov Equations

and Applied Analysis 3 Solving this system, with the aid of Maple, we obtain the following values for the constants: A 0 = −c2]2 + ]2 − 1 ] ,

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.

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 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  1 = / and  2 = /, 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: (5)

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  2 =  2 − 4 > 0 and  being a constant of integration [28].

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 A  , ( = 0, 1, . . ., ) are parameters to be determined.The balancing procedure yields  = 2, 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 A 0 , A 1 , and A 2 : 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.

Solutions of (1) Using Ricatti as the Simplest Equation.
The balancing procedure yields  = 2, 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 A 0 , A 1 , and A 2 .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  2 =  2 − 4 > 0 and  is a constant of integration.

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  =  = 1 and  =  = 1 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

The Klein-Gordon-Zakharov Equations
The Klein-Gordon-Zakharov (KGZ) equations [32]   −   +  + V + || 2  = 0, (22a) are a coupled system of nonlinear partial differential equations of two functions (, ) and V(, ).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 V(, ) denotes the deviation of ion density from its equilibrium.Here (, ) is a complex function and V(, ) is a real function.Note that if we remove the term || 2 , then this system reduces to the classical Klein-Gordon-Zakharov system [33]   −   +  + V = 0, (23a) A number of studies have been conducted for this system ((23a) and ( 23b)) in different time space [34][35][36][37][38].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).

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 sn( | ) is a Jacobian elliptic function of the sine amplitude [40], is the modulus of the elliptic function with 0 <  < 1.Here  1 and  2 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 V(, ) can be obtained from (26).
It should be noted that the solution ( 32) is valid for 0 <  < 1, as  approaches zero, the solution becomes the normal sine function, sin , and as  approaches 1, the solution tends to the tanh function, tanh .
The profile of the solution (32) is given in Figure 1.

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.
where  and  are constants.The balancing procedure yields  = 1, 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.

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  = 1, so the solution of ( 27) is of the form  () =  0 +  1  () .
where  =  +  and  =  + . is given by √  2 − 4,  is a constant of integration, and  0 and  1 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 V(, ).

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.