Optical solitons to the Perturbed Gerdjikov-Ivanov equation with quantic nonlinearity

In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi elliptic function expansion method. This results in obtaining distinct solutions including dark, bright, singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The 2- and 3-dimensional graphs of the reported solutions are presented. The reported results may be useful in explaining the physical features of the studied equation.


Introduction
A nonlinear system is one in which the change in the output is not related to the shift in the input in mathematics and science. So most systems are innately nonlinear, nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists. In contrast to relatively simpler linear systems, nonlinear dynamical systems, which characterize changes in variables over time, may appear chaotic, unpredictable, or inconsistent Korenberg and Hunter (1996), Gintautas (2008). This kind of nonlinear system can be mathematically represented by nonlinear partial and/or ordinary differential equations. A differential equation in mathematics is an equation that specifies a correlation between one or more unknown functions and their derivatives. In practical uses, the functions portray metrics, the derivatives their rates of change, and the differential equation affirms the correlation between the two. Because of the prevalence of these connections, differential equations are extensively utilized in various fields including engineering, physics, economics, and biology Wheeler and Crummett (1987).
As addressing nonlinear dynamical equations is complicated, nonlinear systems are often simulated via linear equations (linearization). Up to a specific accuracy and range for the input values, this operates well, but linearization masks numerous exciting features like solitons, chaos, and singularities. As a result, some parts of the dynamic properties of a nonlinear system seem rather unreasonable, erratic, or even chaotic. Such chaotic behavior may seem to be a random activity, but it is not. For instance, some atmospheric patterns are regarded to be chaotic, whereby little alterations addressed to one aspect of the system cause major alterations to the entire system. Precise protracted predictions are unattainable with the existing technologies owing to this nonlinearity Campbell (2004), Lazard (2009), Mosconi et al. (2008). Various analytical methods have been utilized to solve different kinds of nonlinear equations, such as the extended direct algebraic method Hubert et al. (2018), the extended trial function method Biswas et al. (2018), Ablowitz et al. (1991), the inverse scattering method Anker and Freeman (1978), the Kudryashov expansion and sinecosine method , the Jacobi elliptic function (Tarla et al. 2022a, 2022b, modified generalized exponential rational function ) and many others (Tarla et al. 2022a, Khalili Golmankhaneh et al. 2021, Tarla et al. 2022b, 2022c, Ali and Varol (2019, Ismael et al. 2022a, Manafian et al. 2021, 2022b, 2022c, 2022d, Alshehri et al. 2022b, Ur-Rehman and Ahmad 2022. The Perturbed Gerdjikov-Ivanov equation is given as follows Younis et al. (2021, Muniyappan et al. 2021, Hosseini et al. 2020, Samir et al. 2022, Altwaty 2022 where W(x, t) is the complex-valued function and the dependent variable represents the wave profile. The x and t independent variables are defined as temporal and spatial variables, respectively. The initial term W t in the equation stands for linear temporal evolution and the second term W xx stands for the group velocity dispersion while the third term |W| 4 W stands for the quantic nonlinearity. The symbol stands for the inter-model dispersion and (|W| 2 W) x stands for the self-steepening and (|W| 2 ) x W portrays the nonlinear dispersion. In Muniyappan et al. (2021), the authors used two different methods to investigate dark soliton solutions. In another study Hosseini et al. (2020), the authors obtained solutions using newly well-established methods. There is a difference between the solutions of this study and the solutions we obtained. Because the solution functions of the methods used are different and the method structure is different. It is seen that improved modified extended tanh function method was used in the study by Samir and others Samir et al. (2022). When we compare the graph given in the dark solution of this study with the wave profile of our dark solution, the b quantic nonlinearity term in our study varies according to the a velocity dispersion term.

Method
The main steps of the JEFEM are formally outlined in this section. For this aim, take into account the nonlinear PDE as follows: where, F is a polynomial function within W(x, t) and its partial derivatives. By considering the following wave transformation Assume that the nonlinear PDE mentioned before may be transformed into a NODE as through adopting the transformation here, k 1 , k 2 , and v are constants. The following steps will be taken in order to create exact solutions: Step 1: Suppose that the following solves the NODE: where, g i and f i are constants ( g D ≠ 0 or f D ≠ 0 ). The function z( ) is given by where, s 1 , c 1 and r 1 are constants.
Step 2: We will find the value of D by applying the balance principle which is defined between the non-linear term with the highest-order derivative term in Eq. (3).
Step 3: Using Eq. (3) and Eq. (5) in Eq. (6), a polynomial dependent on z( ) will be provided. Setting the coefficients of same power of z i ( ) , b = 0, 1, 2, ... to zero, we provide a system of algebraic equations.Then, using computer tools, we solve the resulting system to reach the unknown parameters.
Step 4: The solutions of Eq. (5) according to the conditions of s 1 , c 1 and r 1 are as follows (Table 1):  The function z( ) changes to hyperbolic and trigonometric functions according to the values of the Jacobi elliptic functions m, in the cases of m → 0 and m → 1 . Thereby, solutions structures are hyperbolic function solutions when m → 1 as sn ( , m)

Applications
In this section, we utilize the JEFEM to the Pertürbed Gerdjikov-Ivanov equation which is given in Eq. (1).
Taking the wave transformation Eq.
(3) into account, we have We provide the following real parts of Eq. (7) as and the imaginary part is Considering the coefficients of Eq. (9) to zero, one may have v = −2ak 1 − and c = −3 − 2 . If is taken, the ordinary differential equation will be as follows.
guidance, and peakons, whose peaks have a discontinuous first derivative, are examples of such solitary waves. It is identified that dark soliton defines solitary waves with a slower speed than the background, bright soliton defines solitary waves whose peak strength is larger than the background, and singular soliton solutions is a solitary wave with discontinuous derivatives. We consider the results offered in our study may be relevant in illuminating the physical impact of the model investigated. In future work, we can apply the JEFEM to some other nonlinear partial differential equations in the standard form and in the fractional form. Consequently, the solutions obtained are in the form of dark solutions for Eq. (13) and (26) as presented in Figs. 1 and 6, respectively, trigonometric solutions for Eq. (14) as shown in Fig. 2, periodic solutions for Eq. (15) as seen in Fig. 3, bright solutions for Eq. (16) as shown in Fig. 4, and hyperbolic solutions for Eq. (24) as demonstrated in Fig. 5. In addition to, the solutions obtained are in the form of singular solutions for Eq.