On some new analytical solutions for new coupled Konno–Oono equation by the external trial equation method

In this paper, the external trial equation method is employed to solve new coupled Konno–Oono (CKO) equation. By using this method, new exact solutions involving parameters, expressed by generalized hyperbolic and elleptic solutions are obtained. The current method presents a wider applicability for handling nonlinear wave equations. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of CKO equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this soliton wave equation.

Extended trial equation method is one of the robust techniques to look for the exact solutions of nonlinear partial differential equations that has received special interest owing to its fairly great performance. For example, Mohyud-Din and Irshad [44] explored new exact solitary wave solutions of some nonlinear PDEs arising in electronics using the extended trial equation method. Mirzazadeh et al [45] adopted he extended trial equation method to obtain analytical solutions to the generalized resonant dispersive nonlinear Schrödinger's equation with power law nonlinearity. Ekici et al [46] found the exact soliton solutions to magneto-optic waveguides that appear with Kerr, power and log-law nonlinearities using the extended trial equation method. This paper will adopt an integration scheme that is known as the external trial equation method that will reveal soliton solutions as well as other solutions. The system for the model studied here to investigate exact solution structures. We note that this system has not yet been studied using the extended trial equation method.
The rest of the paper is ordered as follows: we present the extended trial equation method in section 2. In sections 3, application of the new coupled KO equation is given and derived exact solutions. The graphical illustrations of some solutions are given in section 4. Finally, the conclusion is given in section 5.

Extended trial equation method
The current method described here is the extended trial equation method utilized to find traveling wave solutions of LPD model which can be understood through the following steps: Step 1. We assume that the given nonlinear PDE  where 0 l ¹ and c 0 ¹ . Substituting equation (2.2) into (2.1) yields a nonlinear ordinary differential equation,

Utilizing the wave transformation
Step 2. Take the transformation and trial equation as follows: Using the equations (2.4) and (2.5), we can find where F G ( ) and Y G ( ) are polynomials. Substituting these terms into equation (2.1) yields an equation of polynomial L G ( ) of Γ: By utilizing the balance principle on (2.8), we can determine a relation of ,  q and δ. We can take some values of ,  q and δ.
Step 3. Setting each coefficient of polynomial L G ( ) to zero to derive a system of algebraic equations: By solving the system (2.9), we will obtain the values of , , , Step 4. In the following step, we obtain the elementary form of the integral by reduction of equation (2.5), as follows where 0 h is an arbitrary constant.

Application of ETEM for coupled KOE
New CKO equation system which is a coupled integrable dispersionless equations is given as follows By the same manipulation as illustrated in the previous section, we can determine values of , , d q and ò, by balancing U 3 and U  in equation (3.6) as follows: For different values of , , d q and ò, we have the following cases: = , and 0  = for equations (2.4) and (2.5), then we obtain where 0 4 x ¹ and 0 0 z ¹ . Solving the algebraic equation system (2.9) yields •First set of parameters: Substituting these results into equations (2.5) and (2.10), we get 3.11 x . Therefore, the solution for the CKO equation will be as Therefore, the solution for the CKO equation will be as 3.17 a t x and 4 x can be selected as free constants.
Third solution: Therefore, the solution for the CKO equation will be as x a x a x x x a x S = + + + ( )and 3 a , 0 x , and 4 x can be selected as free constants. Fourth solution: 3.26 Case II: Then, we obtain the following results as where k , , , , 3 .

Conclusion
In this paper, the new CKO equation was successfully studied. The mentioned task was accomplished by adopting the ETEM to generate a series of exact traveling wave solutions. A comparison of our results with those obtained in [8] by using the sine-Gordon expansion method shows, that there are many new solutions in the present work. Some graphical figures were also portrayed to demonstrate the dynamic behavior of extracted solutions. The observations confirm that above method is efficient algorithms for analytic treatment of a wide range of nonlinear systems of PDEs which arising in nonlinear physics.