Computational and traveling wave analysis of Tzitzéica and Dodd-Bullough-Mikhailov equations: An exact and analytical study

: Computational and travelling wave solutions provide significant improvements in accuracy and characterize novelty of imposed techniques. In this context, computational and travelling wave solutions have been traced out for Tzitzéica (cid:0) (cid:1) and Dodd-Bullough-Mikhailov equations by means of 1/ G (cid:48) -expansion method. The different types of solutions have constructed for Tzitzéica and Dodd-Bullough-Mikhailov equations in hyperbolic form. More-over, solution function of Tzitzéica and Dodd-Bullough-Mikhailov equations has been derived in the format of log-arithmic nature. Since both equations contain exponential terms so the solutions produced are expected to be in log-arithmic form. Traveling wave solutions are presented in different formats from the solutions introduced in the lit-eratur (cid:0) e. (cid:1) The reliability, effectiveness and applicability of the 1/ G (cid:48) -expansion method produced hyperbolic type solutions. For the sake of physical significance, contour graphs, two dimensional and three dimensional graphs have been depicted for stationary wave. Such graphical illustration has been contrasted for stationary wave verses traveling wave solutions. Our graphical comparative analysis suggests that imposed method is reliable and powerful method for obtaining exact solutions of nonlinear evolution equations.


Introduction
Nonlinear evolution equations are studied in many areas such as physics, engineering, social sciences and chemistry. Particularly in the last years, many studies have been conducted to attain either exact or analytical solutions of nonlinear evolution equations. Analytic solutions of nonlinear evolution equations have been investigated and traced out by means of several mathematical methods. For the sake of con rmation and evaluations, Baskonus and Cattani [1,2] invoked extended sinh-gordon equation expansion method. Improved Bernoulli sub-equation function method has been employed by Bulut and Dusunceli [3,4] via fractional and non-fractional appoaches. Maih et al. [5] suggested G G − G -expansion method for closed form traveling wave solutions of integro-di erential equations. Yokus and Durur presented (1+1)-dimensional resonant nonlinear Schrödinger's equation by means of G Gexpansion method in [6][7][8]. Same author worked on sub equation method in [9] and G -expansion method in [10][11][12]. The partial solutions via decomposition method and residual power series method have been found for solitary and periodic solutions for the coupled higherdimensional Burgers equations in [13][14][15][16][17]. Additionally Sumudu transform method is employed in [18]. The numerical study and three-dimensional elliptic partial differential equations have been haar wavelet collocation method [19,20] and rst integral method [21]. The most important advantage of the expansion methods is the generation of traveling wave solutions of partial di erential equations, such solutions are di cult for investigation. The most important common feature of the expansion methods are easy to apply and useful for traditional wave transform. With the help of this transformation, partial di erential equation can be converted into an ordinary di erential equation. In this continuity, we consider the Tzitzéica equation of the form [22]: We consider Dodd-Bullough-Mikhailov equation [22] u xt + e u + e − u = . (2) Equations (1-2) play an important role in the eld of solidstate physics and nonlinear optics. Various studies on equations (1-2) have been carried out by many researchers; for instance, Kumar et al. [23] have explored traveling wave solutions of the Tzitzéica type equations by invoking sine-Gordon expansion method. The new types of exact solutions and class of solitary-like solutions of the Tzitzéica equation have been explored by means of similarity reduction and classical Lie symmetry analysis by Huber [24]. Solitons and periodic solutions have been obtained for equations (1-2) using tan h method in [25] in which analytic solutions for equations (1-2) are obtained using improved tan(ϕ(ξ ) / ). Moreover, recent attempts have been explored on the basis of variety of the solutions as expansion method [26], exponential rational function method [27], expa function method [28], dressing factor method [29]. In brevity, we derived and obtained analytic solutions of equations (1-2) using ( /G )-expansion method. Additionally, the recent attempts of analytical techniques and exact solutions can be traced out [30][31][32][34][35][36][37][38][39][40][41][42][43] and [44][45][46][47] therein.

( /G )-Expansion method
We get the general form of non-linear partial di erential equation in the general form.
where v is a constant and the velocity of the wave. One can convert into following nonlinear ordinary di erential equation (ODE) for U (ξ ): The solution of Eq. (4) is assumed to have the form where a i , (i = , , ..., N) are constants and G = G (ξ ) provides the following second-order integer order di erential equation as where λ and µ are constants to be determined after, where, A is integral constant. Some constraints are needed for the solution of Eq. (7). These restrictions are the ignoring of values that makes the solution unde ned. The asymptotic behavior of the solution to include a singular point is attractive for employees. In addition, the existence of a single point in the solution produced by the method o ers a di erent perspective to the shock wave phenomenon. If the desired derivatives of the Eq. (5) are calculated and substituting in the Eq. (4), a polynomial with the argument G is attained. An algebraic equation is created by equalizing the coe cients of this polynomial to zero. These equations are solved with the help of a package program and put into place in the default Eq.

Application of ( /G )-expansion method . The solution of Tzitzéica equation
We want to get the di erent type of exact solutions for the Eq.
where k, w are real numbers. Replacing Eq. (9) into Eq. (8) for obtaining nonlinear ODE: In Eq. (10), we nd balancing term sayN = and in Eq. (5), solving both the following situation is obtained (11) Replacing Eq. (11) into Eq. (10) and the coe cients of the algebraic Eq. (1) are equal to zero, we can establish the following algebraic equation systems: : − a = , : −k λ a a + w λ a a − a a = , Case1.
In addition, if Eq. (14) is written instead of u = ln v transformation, the analytical solution of Eq. (1) is as follows.
In addition, if Eq. (17) is written instead of u = ln v transformation, the analytical solution of Eq. (1) is as follows.
In addition, if Eq. (20) is written instead of u = ln v transformation, the analytical solution of Eq. (1) is as follows. .

The solution of the Dodd-Bullough-Mikhailov equation
We consider Eq. (2). Using transmutation u = ln v, we get Suppose that solution of Eq. (22) may be write as follows where k, w are real numbers. Replacing Eq. (23) into Eq. (22) to obtain nonlinear ODE: In Eq. (24), we obtain balancing term N = , then the following relationship is achieved: Replacing Eq. (25) into Eq. (24), and the coe cients of the algebraic of Eq. (2) are equal to zero, we may nd the following algebraic equation systems, : kwλ a a + a a = , : kwλµa a + a a + kwλ a a + a a = , : kwµ a a + kwλµa + a + kwλµa a + kwλ a a + a a a = , : kwµ a + kwµ a a + kwλµa a + a a + a a = , : kwµ a a + kwλµa + a a = , : kwµ a + a = .
Considering Eq. (7), substituting Eq. (30) into Eq. (25), the following solution is obtained In addition, if Eq. (31) is written instead of u = ln v transformation, the analytical solution of Eq. (2) is as follows

Results and discussion
One of the important instruments used in obtaining the exact solution of nonlinear evolution equations is the expansion method. In the literature, traveling wave solutions have been obtained by using many expansion methods. The di erent types of traveling wave solutions were constructed from G G -expansion method, novel exponential rational function method, sub equation method, extended auxiliary equation method, the tanh method, Kudryashov method, Sine-Gordon expansion method, the extended tanh method, the im- Considering the Eq. (33), it can be said that Eq. (15) exhibits a damped wave behavior independent of the variable t. Because for x → ±∞ is u → . In addition, we can observe the progress of the following solitary wave for di erent values of t time parameter, x ∈ [ , ] of the traveling wave solution presented by Eq. (15). Figure 6 presents that a wave moves to the right. One of the most important reasons for this is that w > represents the velocity of the wave. w = . that is chosen for Figure 6.

Conclusions
In this article, we have obtained traveling wave solutions for the Tzitzéica and the Dodd-Bullough-Mikhailov equations by invoking of G -expansion method. For both equations, the solutions are obtained in hyperbolic form. The validity of the ( /G )-expansion method has been tested by checking the hyperbolic type solutions. Consequently, nonlinear part of the method can be recommended for generating di erential equations in traveling wave solutions. The 3D, 2D and contour graphics representing the constant waves are presented by evaluating the constants in the traveling wave solutions. This method is an important instrument in obtaining the traveling wave solutions of nonlinear evolution equations. The reliability of this method can be used to obtain the analytical solution of di erential equations in di erent classes. Due to this fact, this method increases the importance of its methodology. To conclude that complex processes and di culties have been overcome with the help of computer programs.