The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

The improved generalized tanh-cothmethod is used in nonlinear sixth-order solitary wave equation.Thismethod is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple.


Introduction
Nonlinear evolution equations (NLEEs) play an important role in various branches of scientific disciplines, such as fluid mechanics, optical fibers, plasma physics, chemical physics, biology, solid state physics, oceans engineering, and many other scientific applications.The solitary wave was introduced by Russell more than a century ago [1].In the past years, many powerful methods for finding exact solutions of NLEEs have been proposed, such as the generalized (  /)-expansion method [2], the tanh-coth method [3], the modified sine-cosine method [4], the generalized unified method [5], the improved -expansion method [6], the generalized Kudryashov method [7], the generalized Riccati equation mapping method [8], the modified Kudryashov method [9], the exp(−()) method [10], the lie symmetry analysis method [11], the first integral method [12], and the consistent Riccati expansion [13].
Another powerful method has been presented by Malfliet [14], who had customized the tanh technique and called the tanh method.In 2002, Fan and Hona [15] extended the tanh method which is called the extended tanh method, by using () = ∑  =0     as traveling wave solutions.In 2007, Wazwaz [3] extended and improved this method which is called the tanh-coth method.In this method () = ∑  =0     + ∑  =1    − is used as traveling wave solutions.In 2008 Gómez and Salas [16] improved and generalized this method which is called the improved generalized tanh-coth method, by using () = ∑  =0   ()  + ∑ 2 =+1   () − , where  is the solution of the generalized Riccati equation.Afterwards, several researchers applied this method to obtain new exact solutions for nonlinear PDEs [17][18][19][20].
In 2017, Christou [21] studies solitons occurring in electrical nonlinear transmission lines; there are called electrical solitons.The problem is applied to Ohm's law of solid state physics by using Taylor-series expansions.
In this paper, we focus on using the improved generalized tanh-coth method for finding exact solutions of the sixthorder solitary wave equation: which was proposed by Christou [21] and In Section 2, we briefly describe the improved generalized tanh-coth method; in Section 3, the improved generalized tanh-coth method is applied to the sixth-order solitary wave equations.The last section is short summary and discussion.
The traveling wave transformation is given by where  is the wave speed.We can reduce (3) to the ordinary differential equation According to the improved generalized tanh-coth method, we seek the exact solution of (3) that can be expressed in the following form: where  is a positive integer that will be determined by balancing the highest order derivative term with the highest order nonlinear term.The coefficients   are constants (  ̸ = 0 and  − ̸ = 0) that are determined later while the new variable () is the solution to the generalized Riccati equation where , , and  are constants.The solutions of generalized Riccati equation are given by [18].

The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation
We use the wave transformations (, ) = (),  =  −  +  0 , to reduce (1) to the following ODE: Balancing the highest order term   with the highest order nonlinear term ( 3 )  in (13), we have then  = 2. Consequently, we set Using ( 6) and ( 14) in ( 13) and equating all the coefficients of power of () to be zero, we obtain a system of algebraic equations in the unknowns  0 ,  1 ,  2 ,  3 ,  4 , , , , and .
Solving the system of algebraic equations with the aid of Maple, using (18), we obtain the following results.