Modeling of Dark Solitons for Nonlinear Longitudinal Wave Equation in a Magneto-Electro-Elastic Circular Rod

In this paper, sub equation and 1=G’ ð Þ expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.


Introduction
Nonlinear evolution equations (NLEEs) are used in various fields such as biological sciences, plasma physics, quantum mechanics, fluid dynamics and engineering. Many methods have been used to obtain solutions of NLEEs from past to present.
In particular, 1=G' ð ÞÀexpansion method that we will consider in this study produces hyperbolic type traveling wave solution, while the sub equation method produces dark solitons. Generally, dark solitons are solutions that contain tangent function.
We know that solitons have an important place in wave theory. There are many solitons that offer a mathematical perspective to many physical phenomena. Some of these are dark soliton, bright soliton, peaked solitary, topological soliton, non-topological soliton, singular soliton and so on. The mathematical expressions of these solitons appear as a solution of NLEEs.
It is very difficult to obtain the analytical solution of NLEEs. However, with the help of classical wave transformation, traveling wave solutions can be obtained by converting to ordinary differential equations.
Let's take the LWE in a magneto-electro-elastic circular rod [39] where g is the linear longitudinal wave velocity and q is the dispersion parameter for a magneto-electroelastic circular rod, all of which depend on the material property and geometry [40].
The real-world physical response of a magneto-electro-elastic circular rod LWE is the combination of piezomagnetic and piezoelectric BaTiO 3 [39]. The solutions offered especially for those working in this field are important. It will become more important with the physical meaning of the constants in the solution.
In this study, some researchers have examined the physically precious LWE. In the study of Iqbal et al. wave solutions have been obtained with extended auxiliary equation mapping and extended direct algebraic mapping methods [39]. Ilhan et al. have provided solutions including complex, hyperbolic and trigonometric functions with sine-Gordon expansion method [40]. Baskonus et al. have been presented topological, non-topological and singular soliton solutions using the extended sinh-Gordon equation expansion method [41]. Also, Baskonus et al. have obtained hyperbolic, complex and complex hyperbolic function solutions with the modified exp expansion function method [42]. In their study, Yang et al. achieved solitary wave solutions that peaked using direct integration with the boundary condition and symmetry condition [43]. Younis et al. have been presented dark, bright and singular solitons solutions with the solitary wave ansatz method [44].
In this study, we will present different solutions from the solutions presented in the literature. In particular, we offer a different solution than the dark solitons that Younis et al. present in their work.

Sub-Equation Method
Consider the sub-equation method for the solving NLEEs. Regard the NLEEs as Applying the wave transmutation Eq.
(2) converts into ODE where w is arbitrary constant. In the form it is supposed that Eq. (4) has a solution U n ð Þ ¼ X n i¼0 a i f i n ð Þ; a n 6 ¼ 0; in here a i ; 0 i n ð Þare constants to be determined, n is a positive integer value which is going to be attained in Eq. (4) by balancing term is found according to the principle of balance and the solution of where l is an arbitrary constant. Some exclusive solutions are given of the Riccati equation in (6) as follows: In Eq. (4), if we apply the Eqs. (6) and (5), we attained the new polynomial with respect f n ð Þ a nonlinear algebraic equation system in a i ; i ¼ 0; 1; …; n ð Þ setting all the coefficients of to zero yields f i n ð Þ; i ¼ 0; 1; …; n ð Þ : To find solutions in nonlinear algebraic equations to we determine constants l; s; k; r; a i ; i ¼ 0; 1; …; n ð Þ : Substituting attained constants from this system and by the aid of the formulas (7) the solutions of Eq. (6) into Eq. (5). Then, we obtain analytic solutions for Eq. (2).
Using this analytical method, trigonometric provides solutions of hyperbolic and algebraic type. These solutions are in Eq. (7) formats. Especially our tanh solution contains dark soliton feature [45]. This method is a reliable, effective and powerful analytical method in obtaining the analytical solution of many differential equations.

The (1/G')-Expansion Method
Consider a general form of NLEEs, u; @u @t ; @u @x ; @ 2 u @x 2 ; … ¼ 0: where w is a constant and the speed of the wave. We can convert it into the following nODE for U n ð Þ: The solution of Eq. (9) is assumed to have the form where a i ; i ¼ 0; 1; …; n are constants, n is the balancing term that we need to calculate based on the homogeneous balance principle. G ¼ G n ð Þ provides the following second order IODE: where and d are constants to be determined after, where B is integral constant.
After calculating the n balancing term, the structure of the solution function of the assumed Eq. (10) emerges. The necessary derivatives of this solution are taken and replaced in the Eq. (9), and after some algebraic operations, a polynomial that accepts the expression 1=G 0 ð Þ i ; i ¼ 0; 1; 2; …; n ð Þas a variable can be created. Considering the zero polynomial property, the coefficients of the variable are equal to zero and an algebraic system of equations is obtained. We can reach the solution of the algebraic equation system using ready-made package programs. These solutions are the coefficients of the solution function of the default Eq. (10). When these coefficients are replaced in Eq. (10), there is a solution of Eq. (9). Finally, the classical wave transformation is reversed and the solution of Eq. (8) is reached.
Case 1: If l < 0; Substituting values (18) into (16), we can also present the dark soliton for Eq. (1) using the classical wave transformation inverse, that is, using the n ¼ kx þ wt, as follows: Case II: If l < 0; Substituting values (20) into (16), we can also present the singular for Eq. (1) using the classical wave transformation inverse, that is, using the n ¼ kx þ wt, as follows:  Case III: If l > 0; Substituting values (22) into (16), we attain trigonometric soliton for Eq. (1) Case IV: If l > 0; Substituting values (24) into (16), we attain trigonometric soliton for Eq. (1) Figure 3: 3D, 2D and contour graphs respectively for w ¼ 0:5; g ¼ 2; q ¼ 2; k ¼ 0:2 values of Eq. (23) Figure 4: 3D, 2D and contour graphs respectively for w ¼ 0:5; g ¼ 0:5; q ¼ 2; k ¼ 2 values of Eq. (25) Substituting values (26) into (16), we attain rational traveling wave solution for Eq. (1) 5 Application of (1/G')-Expansion Method Considering Eq. (15), we get balancing term n ¼ 2 and in Eq. (10), the following situation is obtained: ; a 2 6 ¼ 0: Replacing Eq. (28) into Eq. (15) and the coefficients of the algebraic Eq. (1) are equal to zero, can find the following algebraic equation systems: Figure 5: 3D, 2D and contour graphs respectively for r ¼ 0: Replacing values Eq. (30) into Eq. (28) and we have the following hyperbolic type solutions for Eq. (1): 6 Results and Discussion In this study, the LWE of a magneto-electro-elastic circular rod has been successfully produced using two different analytical methods. These solutions are emphasized to be hyperbolic, rational, dark soliton and trigonometric type traveling wave solutions. Generating the solutions of this equation is mathematically valuable as much in terms of physical meaning. The u solutions presented in this article represent the electrostatic potential of the magneto-electro-elastic circular rod. Also, using this potential, a different physical perspective can be presented. If we represent the pressure of this physical event with P, the P pressure can be calculated for different analytical solutions as follows: The potential u calculated here and the P pressure magneto-electro-elastic circular rod can offer different interpretations and different perspectives on the physical event [46][47][48][49].
The figures presented in this work are the graphs of solitons representing the standing wave. Fig. 1 represents the dark soliton, Fig. 2 represents the singular soliton, Figs. 3 and 4 represent the trigonometric solitons, Fig. 5 represents the rational soliton, and Fig. 6 represents the graphics of hyperbolic type solitons.

Conclusions
In this study, we have achieved hyperbolic, rational, dark soliton and trigonometric traveling wave solutions for the LWE in a magneto-electro-elastic circular rod using sub equation method and ð1=G'Þ-expansion method. By giving arbitrary values to the constants in the solution obtained, 3D, 2D and contour graphics of the solution representing the stationary wave are presented. It has been observed that the methods used are easy, effective and powerful, and solutions of NLEEs can be obtained. It would be even more valuable to add a physical meaning to these solutions in the future. Computer package program was used in the construction of these solutions.