Dynamic Properties of Non-Autonomous Femtosecond Waves Modeled by the Generalized Derivative NLSE with Variable Coefficients

Abstract

The primary purpose of this study is to analyze non-autonomous femtosecond waves with various geometrical configurations correlated to the generalized derivative nonlinear Shrödinger equation (NLSE) with variable coefficients. Numerous academic publications, especially in nonlinear optics, material science, semiconductor, chemical engineering, and many other fields, have looked into this model since it is closer to real-world situations and has more complex wave structures than models with constant coefficients. It can serve as a reflection for the slowly altering inhomogeneities, non-uniformities, and forces acting on boundaries. New complex wave solutions in two different categories are proposed: implicit and elliptic (or periodic or hyperbolic) forms are obtained for this model via the unified method. Indeed, the innovative wave solutions that were achieved and reported here are helpful for investigating optical communication applications as well as the transmission characteristics of light pulses.

1. Introduction

Studies on optical wave solutions through optical fiber media that demonstrate quintic nonlinearity and self-steepening influence have become vigorous, continuous and are motivated by their important applications, owing to their ability of propagating long distances without wane and due to their various geometrical structures, to high-capacity fiber telecommunications, and to all optical switches [1,2,3,4,5]. In many aspects of mathematics and science, including nonlinear optics, hydrodynamics, quantum physics, nonlinear acoustics, and many others, the nonlinear Schrödinger equation (NLSE) and its relatives have an essential role in this aspect [6,7,8,9,10,11,12]. The NLSE is the most essential equation in chemistry since it addresses the quantum-mechanical interaction of particles [13]. Therefore, it is important to find precise solutions for a wide variety of compounds and geometries when developing new substances such as medications or catalysts.

The NLSE’s dispersion and nonlinear coefficients, which depict the frequency curvature versus wave number dispersion and the carrier frequency change with signal amplitude, play a crucial role in finding its analytical solutions [14]. The NLSE promotes soliton solutions that can be used in electronic circuits, ultrafast soliton switches, soliton lasers, and optoelectronic devices [15,16,17]. Regarding nonlinear optics, non-Kerr and Kerr nonlinearity have received much attention [18]. The research field of ultrafast events in nonlinear optical fibers has been analyzed by higher-order NLSE, along with order three dispersion, self-steepness, self-frequency shift, and other phenomena [19,20,21].

Many different approaches have been proposed in the effort to tackle the enormous variety of distinct forms of NLSE. For the NLSE with arbitrary dual-power law parameters, Kudryashov and Biswas [22] employed the method of transformation of dependent and independent variables to establish solitary wave solutions. The higher dimension NLSE that occurred in the regions of irregular dispersion was analyzed via the extended Riccati equation mapping scheme in [23]. Additionally, the authors found approximations to some of the obtained solutions, among them the bright and dark optical solitons, using the q-homotopy algorithm coupled with the Laplace transform. In nonlinear optical fibers, bright and dark optical solitons of the (2 + 1)-dimensional perturbed NLSE were researched and addressed by Wazwaz [24]. In the recent past, unstable NLSE was solved using the modified extended mapping technique [25]. In [26], the (G’/G)-expansion strategy was implemented to acquire the specific wave solutions of the equation. A simple equation strategy for forming the results of unstable NLS equations was elaborated by Lu et al. in [27].

This paper demonstrates new types of wave solutions for the generalized derivative NLSE with varying coefficients (Vc-GDNLSE) [28] via an efficient technique, namely the unified method [29,30]. In this case, when the nonlinearity parameter and dispersion coefficient are treated as variable functions in optical communication, the NLSE model with variable coefficients simulates significant phenomena [31]. The Vc-GDNLSE has the following structure [28].

$$\begin{array}{c}\hfill i{\mathrm{\Gamma }}_z={\rho }_0\left(z\right){\mathrm{\Gamma }}_{tt}+{\rho }_1{\left(z\right)\left|\mathrm{\Gamma }\right|}^2\mathrm{\Gamma }+{\rho }_2\left(z\right){\left|\mathrm{\Gamma }\right|}^4\mathrm{\Gamma }+i{\left[({\hslash }_0\left(z\right)+{\hslash }_1\left(z\right)|\mathrm{\Gamma }{|}^2)\mathrm{\Gamma }\right]}_t,\end{array}$$

(1)

where ${\rho }_0\left(z\right),{\rho }_1\left(z\right),{\rho }_2\left(z\right)$, ${\hslash }_0\left(z\right)$, and ${\hslash }_1\left(z\right)$ are arbitrary functions in z and $\mathrm{\Gamma }=\mathrm{\Gamma }(t,z)$ is a complex-valued function that symbolizes the electric field. Equation (1) is commonly utilized when its coefficients are constants, in other words, it has the construction stated underneath

$$\begin{array}{c}\hfill i{\mathrm{\Gamma }}_z={\rho }_0{\mathrm{\Gamma }}_{tt}+{\rho }_1{\left|\mathrm{\Gamma }\right|}^2\mathrm{\Gamma }+{\rho }_2{\left|\mathrm{\Gamma }\right|}^4\mathrm{\Gamma }+i{\left[\right({\hslash }_0+{\hslash }_1\left|\mathrm{\Gamma }{|}^2\right)\mathrm{\Gamma }]}_t,\end{array}$$

(2)

where ${\rho }_0$, ${\rho }_1$, and ${\rho }_2$ denote sequentially the group velocity dispersion, the cubic nonlinearity, and the quintic parameters. Additionally, the two parameters ${\hslash }_0$ and ${\hslash }_0$ are progressively taken into account as the group velocity and the self-steepening effect. This model, which takes higher-order effects into account, is important in determining how femtosecond waves behave in optical fibers. Equation (2) becomes the Kaup–Newell equation, which characterizes how the polarized nonlinear Alfvén waves spread in plasmas when ${\rho }_1={\rho }_2=0$ [32]. While it represents the Chen–Lee–Liu equation, which describes how cubic and quadratic nonlinearities interact to predict short-pulse spread in a frequency-doubling crystal when ${\rho }_1={\rho }_2={\hslash }_1=0$ [33].

The layout of the article is divided into the following three sections: the construction of the unified technique is covered in Section 2. The application of the aforesaid approach to solve our problem is described in Section 3, and the discussion of the dynamic properties of the produced solutions is explored through different figures. Finally, the conclusions are covered in Section 4.

2. An Overview of the Applied Method

In this section, we provide the following nonlinear differential equation with variable coefficients (Vc-NDE) to illustrate the unified method’s algorithm. Suppose we have a Vc-NDE provided by

$$\mathrm{\Pi }(\mathrm{\Gamma },{\mathrm{\Gamma }}_z,{\mathrm{\Gamma }}_t,{\mathrm{\Gamma }}_{zz},{\mathrm{\Gamma }}_{tt},\cdots )=0,$$

(3)

where $\mathrm{\Pi }$ is a polynomial function in its arguments given by Equation (1).

We implement the wave transformation $\xi =t+{\int }_0^{z}q\left(z\right)dz,\mathrm{\Gamma }(t,z)=\mathrm{\Xi }\left(\xi \right)$ into (3). Thus, (3) takes the new form

$$\tilde{\mathrm{\Pi }}(\mathrm{\Xi },{\mathrm{\Xi }}^{\prime },{\mathrm{\Xi }}^{″},\cdots )=0,$$

(4)

where $q\left(z\right)$ is an arbitrary function, $\mathrm{\Xi }=\mathrm{\Xi }\left(\xi \right)$ and ${\mathrm{\Xi }}^{\prime }=\frac{d\mathrm{\Xi }}{d\xi }$ [34,35].

For Equation (4), the unified method divides its outcomes into two groups: polynomial and rational structures, respectively given by [29,30]

$${\displaystyle \mathrm{\Xi }\left(\xi \right)=\sum _{ı=0}^n{a}_{ı}\left(z\right){\left(\mathrm{\Delta }\left(\xi \right)\right)}^{ı},}$$

(5)

and

$${\displaystyle \mathrm{\Xi }\left(\xi \right)=\sum _{ı=0}^n{d}_{ı}\left(z\right){\left(\mathrm{\Delta }\left(\xi \right)\right)}^{ı}/\sum _{\ell =0}^r{h}_{\ell }\left(z\right){\left(\mathrm{\Delta }\left(\xi \right)\right)}^{\ell },n\ge r,}$$

(6)

where $a_{ı}\left(z\right)$, $d_{ı}\left(z\right)$, and $h_{\ell }\left(z\right)$ are free functions in z and $\vartheta \left(\eta \right)$ verifies the next auxiliary equation

$${\displaystyle {\left({\mathrm{\Delta }}^{\prime }\left(\xi \right)\right)}^{\varrho }=\sum _{j=1}^{\varrho k}{b}_j\left(z\right){\left(\mathrm{\Delta }\left(\xi \right)\right)}^j,\varrho =1,2,}$$

(7)

where $b_j\left(z\right)$ are functions in z. For $\varrho =1$ in Equation (7), the obtained solutions are classified into implicit or elementary ones and for $\varrho =2$, we obtain periodic or elliptic solutions (more discussions are found in [29,30]). Herein, we restrict our discussion to the polynomial-type solutions for Equation (2). In contrast, the other solutions of this equation will be studied in a future work in relation to the conformable derivative.

3. Complex Wave Solutions for Equation (1) via the Unified Method

This section contains several complex wave solutions with various geometrical patterns for the model specified by Equation (1). First, we insert the new transformation $\mathrm{\Gamma }(t,z)=\mathrm{\Xi }\left(\xi \right)\exp \left(i(\eta +\chi (\xi \left)\right)\right),\xi =t+{\int }_0^{z}q\left(z\right)dz$ and $\eta =\kappa t+\theta z$ into Equation (1) which yields two different ODE:

$$\begin{array}{ccc}\hfill & & {\rho }_0\left(z\right){\mathrm{\Xi }}^{″}\left(\xi \right)+\mathrm{\Xi }\left(\xi \right)\left(\theta +q\left(z\right){\chi }^{\prime }\left(\xi \right)-{\rho }_0\left(z\right){\left(\kappa +{\chi }^{\prime }\left(\xi \right)\right)}^2-{\hslash }_0\left(z\right)\left(\kappa +{\chi }^{\prime }\left(\xi \right)\right)\right)+\mathrm{\Xi }{\left(\xi \right)}^5{\rho }_2\left(z\right)\hfill \\ & & +\mathrm{\Xi }{\left(\xi \right)}^3\left({\rho }_1\left(z\right)-{\hslash }_1\left(z\right)\left(\kappa +{\chi }^{\prime }\left(\xi \right)\right)\right)=0,\hfill \end{array}$$

(8)

and

$$\begin{array}{c}\hfill -q\left(z\right){\mathrm{\Xi }}^{\prime }\left(\xi \right)+{\rho }_0\left(z\right)\left(2{\mathrm{\Xi }}^{\prime }\left(\xi \right)\left(\kappa +{\chi }^{\prime }\left(\xi \right)\right)+\mathrm{\Xi }\left(\xi \right){\chi }^{″}\left(\xi \right)\right)+{\mathrm{\Xi }}^{\prime }\left(\xi \right)\left(3\mathrm{\Xi }{\left(\xi \right)}^2{\hslash }_1\left(z\right)+{\hslash }_0\left(z\right)\right)=0,\end{array}$$

(9)

wherein ${(.)}^{{}^{\prime }}=\frac{d(.)}{d\xi }$. Multiplying Equation (9) by $\mathrm{\Xi }\left(\xi \right)$ and integrating it with respect to $\xi$, we obtain the relation

$${\chi }^{\prime }\left(\xi \right)=-\frac{-2q\left(z\right)+4\kappa {\rho }_0\left(z\right)+3\mathrm{\Xi }{\left(\xi \right)}^2{\hslash }_1\left(z\right)+2{\hslash }_0\left(z\right)}{4{\rho }_0\left(z\right)},$$

(10)

where the integration constant equals zero. Again, utilizing Equation (10) into Equation (8) reveals a new ODE in the form

$$\begin{array}{ccc}\hfill & -& 8\mathrm{\Xi }\left(\xi \right)q\left(z\right)\left(2\kappa {\rho }_0\left(z\right)+\mathrm{\Xi }{\left(\xi \right)}^2{\hslash }_1\left(z\right)+{\hslash }_0\left(z\right)\right)+4\mathrm{\Xi }\left(\xi \right)q{\left(z\right)}^2+4\mathrm{\Xi }\left(\xi \right)\left(4\theta {\rho }_0\left(z\right)+{\hslash }_0\left(z\right){}^2\right)\hfill \\ & +& 16{\rho }_0\left(z\right){}^2{\mathrm{\Xi }}^{″}\left(\xi \right)+\mathrm{\Xi }{\left(\xi \right)}^5\left(16{\rho }_0\left(z\right){\rho }_2\left(z\right)+3{\hslash }_1\left(z\right){}^2\right)+8\mathrm{\Xi }{\left(\xi \right)}^3\left(2{\rho }_0\left(z\right){\rho }_1\left(z\right)+{\hslash }_0\left(z\right){\hslash }_1\left(z\right)\right)=0.\hfill \end{array}$$

(11)

Multiplying Equation (11) by ${\mathrm{\Xi }}^{\prime }\left(\xi \right)$ and integrating both sides with respect to $\xi$ in the presence of changing variable $\mathrm{\Xi }{\left(\xi \right)}^2=Y\left(\xi \right)$, we obtain

$$Y^{\prime }{\left(\xi \right)}^2+4A\left(z\right)Y{\left(\xi \right)}^2+4B\left(z\right)Y{\left(\xi \right)}^3+4H\left(z\right)Y{\left(\xi \right)}^4=0,$$

(12)

where $A\left(z\right)={\displaystyle \frac{-2q\left(z\right)\left(2\kappa {\rho }_0\left(z\right)+{\hslash }_0\left(z\right)\right)+q{\left(z\right)}^2+4\theta {\rho }_0\left(z\right)+{\hslash }_0\left(z\right){}^2}{4{\rho }_0\left(z\right){}^2}}$, $H\left(z\right)={\displaystyle \frac{16{\rho }_0\left(z\right){\rho }_2\left(z\right)+3{\hslash }_1\left(z\right){}^2}{48{\rho }_0\left(z\right){}^2}}$, and $B\left(z\right)={\displaystyle \frac{{\hslash }_1\left(z\right)\left({\hslash }_0\left(z\right)-q\left(z\right)\right)+2{\rho }_0\left(z\right){\rho }_1\left(z\right)}{4{\rho }_0\left(z\right){}^2}}$ and the integration constant is considered to be zero.

We assume that Equation (5) is a general solution of Equation (12). After using the balancing principle in [29,30], we get $n=k-1$ subject to $1<k\le 3$.

Category I.$\varrho =1$ and $k=2$ or $k=3$

Geometrical 1 ($k=2)$. Solution with solitary wave form.

Clearly, from the relation $n=k-1$, we get $n=1$ and the solution is formulated as

$$\mathrm{\Xi }\left(\xi \right)=a_0\left(z\right)+a_1\left(z\right)\mathrm{\Delta }\left(\xi \right),$$

(13)

where $\mathrm{\Delta }\left(\xi \right)$ verifies Equation (7) as

$${\mathrm{\Delta }}^{\prime }\left(\xi \right)=b_0\left(z\right)+b_1\left(z\right)\mathrm{\Delta }\left(\xi \right)+b_2\left(z\right){\mathrm{\Delta }}^2\left(\xi \right).$$

(14)

Substituting Equations (13) and (14) into Equation (12) and setting all coefficients of ${\mathrm{\Delta }}^{ı}\left(\xi \right),ı=0,1,2$ identical zero, a system of algebraic equations is constructed that can be solved as follows:

$$a_0\left(z\right)=-{\displaystyle \frac{B\left(z\right)-b_1\left(z\right)\sqrt{-H\left(z\right)}}{4H\left(z\right)}},a_1\left(z\right)=-{\displaystyle \frac{b_2\left(z\right)}{2\sqrt{-H\left(z\right)}}},b_0\left(z\right)={\displaystyle \frac{b_1\left(z\right){}^{2}H\left(z\right)+B{\left(z\right)}^2}{4b_2\left(z\right)H\left(z\right)}},A\left(z\right)={\displaystyle \frac{B{\left(z\right)}^2}{4H\left(z\right)}},$$

(15)

where $H\left(z\right)<0$. Thus, Equation (1) has the general solution:

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_1(t,z)& =& \frac{1}{2}\sqrt{-\frac{B\left(z\right)\left(\tanh \left(\frac{\xi B\left(z\right)}{2\sqrt{-H\left(z\right)}}\right)+1\right)}{H\left(z\right)}}\hfill \\ & \times & \exp \left(i\left(\frac{3{\hslash }_1\left(z\right)\left(2\sqrt{-H\left(z\right)}\log \left(\cosh \left(\frac{\xi B\left(z\right)}{2\sqrt{-H\left(z\right)}}\right)\right)+\xi B\left(z\right)\right)+\xi R\left(z\right)}{16H\left(z\right){\rho }_0\left(z\right)}+\eta \right)\right),\hfill \end{array}$$

(16)

where $\eta =\kappa t+\theta z$, $\xi =t+{\int }_0^{z}q\left(z\right)dz$, $R\left(z\right)=8H\left(z\right)\left(q\left(z\right)-2\kappa {\rho }_0\left(z\right)-{\hslash }_0\left(z\right)\right)$, and $H\left(z\right)<0$.

The absolute, the real part, the complex part of the solution given by (16) are depicted in Figure 1.

Geometrical pattern 2 ($k=3)$. Solution with exponential function form.

Again, we get $n=2$ and the solution is formulated as

$$\mathrm{\Xi }\left(\xi \right)=a_0\left(z\right)+a_1\left(z\right)\mathrm{\Delta }\left(\xi \right)+a_2\left(z\right){\mathrm{\Delta }}^2\left(\xi \right),$$

(17)

where $\mathrm{\Delta }\left(\xi \right)$ verifies Equation (7) as

$${\mathrm{\Delta }}^{\prime }\left(\xi \right)=b_0\left(z\right)+b_1\left(z\right)\mathrm{\Delta }\left(\xi \right)+b_2\left(z\right){\mathrm{\Delta }}^2\left(\xi \right)+b_3\left(z\right){\mathrm{\Delta }}^3\left(\xi \right).$$

(18)

Substituting Equations (17) and (18) into Equation (12) and setting all coefficients of ${\mathrm{\Delta }}^{ı}\left(\xi \right),ı=0,1,2,3$ identical zero, a system of algebraic equations is constructed and gives the next solutions:

$$\begin{array}{c}\hfill a_0\left(z\right)=\frac{b_2\left(z\right){}^2}{9b_3\left(z\right)\sqrt{-H\left(z\right)}},a_1\left(z\right)=\frac{2b_2\left(z\right)}{3\sqrt{-H\left(z\right)}},a_2\left(z\right)=\frac{b_3\left(z\right)}{\sqrt{-H\left(z\right)}},A\left(z\right)=\frac{B{\left(z\right)}^2}{4H\left(z\right)},\\ \hfill b_0\left(z\right)=\frac{9b_3\left(z\right)b_2\left(z\right)B\left(z\right)\sqrt{-H\left(z\right)}+2b_2\left(z\right){}^{3}H\left(z\right)}{54b_3\left(z\right){}^{2}H\left(z\right)},b_1\left(z\right)=\frac{b_2\left(z\right){}^2}{3b_3\left(z\right)}-\frac{B\left(z\right)}{2\sqrt{-H\left(z\right)}},\end{array}$$

(19)

where $H\left(z\right)<0$. Consequently, we get the general solution for Equation (1) as

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_2(t,z)& =& \sqrt{\frac{B\left(z\right)}{e^{\frac{\xi B\left(z\right)}{\sqrt{-H\left(z\right)}}}-2H\left(z\right)}}{\cosh }^{-\frac{3{\hslash }_1\left(z\right)}{8\sqrt{H\left(z\right)}{\rho }_0\left(z\right)}}\left(\frac{\xi B\left(z\right)}{2\sqrt{-H\left(z\right)}}\right)\hfill \\ & \times & \exp \left(\frac{i\left(3\xi B\left(z\right){\hslash }_1\left(z\right)+8H\left(z\right)\left(\xi q\left(z\right)+2{\rho }_0\left(z\right)(\eta -\kappa \xi )-\xi {\hslash }_0\left(z\right)\right)\right)}{16H\left(z\right){\rho }_0\left(z\right)}\right),\hfill \end{array}$$

(20)

where $\eta =\kappa t+\theta z$, $\xi =t+{\int }_0^{z}q\left(z\right)dz$, and $H\left(z\right)<0$. The absolute, the real part, the complex part of the solution given by (20) are depicted in Figure 2.

Category II.$\varrho =2$ and $k=2$

Here, we find solutions in elliptic function form. To this end, we suppose that

$$\mathrm{\Xi }\left(\xi \right)=a_0\left(z\right)+a_1\left(z\right)\mathrm{\Delta }\left(\xi \right),$$

(21)

where $\mathrm{\Delta }\left(\xi \right)$ verifies Equation (7) as

$${\mathrm{\Delta }}^{\prime }\left(\xi \right)=\sqrt{b_0\left(z\right)+b_1\left(z\right)\mathrm{\Delta }\left(\xi \right)+b_2\left(z\right){\mathrm{\Delta }}^2\left(\xi \right)+b_3\left(z\right){\mathrm{\Delta }}^3\left(\xi \right)+b_4\left(z\right){\mathrm{\Delta }}^4\left(\xi \right)}.$$

(22)

Substituting Equations (21) and (22) into Equation (12) and setting all coefficients of ${\mathrm{\Delta }}^{ı}\left(\xi \right),ı=0,1,2,3,4$ identical zero, a system of algebraic equations is constructed that can be solved as follows:

$$\begin{array}{ccc}\hfill & & b_1\left(z\right)=\frac{-64A\left(z\right)b_4\left(z\right){}^{3/2}B\left(z\right){(-H\left(z\right))}^{3/2}+4b_3\left(z\right)b_4\left(z\right)H\left(z\right)\left(3B{\left(z\right)}^2-8A\left(z\right)H\left(z\right)\right)+b_3\left(z\right){}^{3}H{\left(z\right)}^2}{16b_4\left(z\right){}^{2}H{\left(z\right)}^2}\hfill \\ & & -\frac{16b_4\left(z\right){}^{3/2}B{\left(z\right)}^3\sqrt{-H\left(z\right)}}{16b_4\left(z\right){}^{2}H{\left(z\right)}^2},a_0\left(z\right)=-\frac{2\sqrt{b_4\left(z\right)}B\left(z\right)+b_3\left(z\right)\sqrt{-H\left(z\right)}}{8\sqrt{b_4\left(z\right)}H\left(z\right)},\hfill \\ & & b_2\left(z\right)=\frac{3}{8}\left(\frac{b_3\left(z\right){}^2}{b_4\left(z\right)}+\frac{4B{\left(z\right)}^2}{H\left(z\right)}\right)-4A\left(z\right),a_1\left(z\right)=\frac{\sqrt{b_4\left(z\right)}}{2\sqrt{-H\left(z\right)}},\hfill \\ & & b_0\left(z\right)=\frac{8b_4\left(z\right)b_3\left(z\right){}^{2}H\left(z\right)\left(3B{\left(z\right)}^2-8A\left(z\right)H\left(z\right)\right)+16b_4\left(z\right){}^{2}B{\left(z\right)}^2\left(16A\left(z\right)H\left(z\right)-3B{\left(z\right)}^2\right)}{256b_4\left(z\right){}^{3}H{\left(z\right)}^2}\hfill \\ & & -\frac{256A\left(z\right)b_3\left(z\right)b_4\left(z\right){}^{3/2}B\left(z\right){(-H\left(z\right))}^{3/2}+64b_3\left(z\right)b_4\left(z\right){}^{3/2}B{\left(z\right)}^3\sqrt{-H\left(z\right)}-b_3\left(z\right){}^{4}H{\left(z\right)}^2}{256b_4\left(z\right){}^{3}H{\left(z\right)}^2}\hfill \end{array}$$

(23)

where $H\left(z\right)<0$ and $b_4\left(z\right)>0$. After that, the general solution of Equation (1) is given by

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_3(t,z)& =& \sqrt{-\frac{\frac{\sqrt{-H\left(z\right)}\left(4b_4\left(z\right)\mathrm{\Delta }\left(\xi \right)+b_3\left(z\right)\right)}{\sqrt{b_4\left(z\right)}}+2B\left(z\right)}{8H\left(z\right)}}{\cosh }^{-\frac{3{\hslash }_1\left(z\right)}{8\sqrt{H\left(z\right)}{\rho }_0\left(z\right)}}\left(\frac{\xi B\left(z\right)}{2\sqrt{-H\left(z\right)}}\right)\hfill \\ & \times & \exp \left(\frac{i\left(3\xi B\left(z\right){\hslash }_1\left(z\right)+8H\left(z\right)\left(\xi q\left(z\right)+2{\rho }_0\left(z\right)(\eta -\kappa \xi )-\xi {\hslash }_0\left(z\right)\right)\right)}{16H\left(z\right){\rho }_0\left(z\right)}\right),\hfill \end{array}$$

(24)

where $\eta =\kappa t+\theta z$, $\xi =t+{\int }_0^{z}q\left(z\right)dz$, $H\left(z\right)<0$ and $\mathrm{\Delta }\left(\xi \right)$ satisfies Equation (22).

It is worth noting that Equation (22) gives different possible solutions of the Jacobi elliptic function type that are related to the choices of $b_{ı}\left(z\right),ı=0,1,2,3,4$.

For a special case when $b_{ı}\left(z\right)=b_{ı},ı=0,1,2,3,4$ are constants, Equation (22) gives a variety of Jacobi elliptic function solutions (JEFS) stated in

Table 1. JEFS with different types $0<\Im <1$.

No.	${\mathit{b}}_0$	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	$\mathbf{\Delta }\left(\mathit{\xi }\right)$
1	1	0	$-({\Im }^2+1)$	0	${\Im }^2$	sn ($\xi$)
2	${\Im }^2$	0	$-({\Im }^2+1)$	0	1	ns ($\xi$)
3	$1-{\Im }^2$	0	$2-{\Im }^2$	0	1	cs ($\xi$)
4	1	0	$2-4{\Im }^2$	0	1	sn ($\xi$) dn ($\xi$)/cn ($\xi$)
5	$\frac{1}{4}(1-{\Im }^2)$	0	$\frac{1}{2}(1+{\Im }^2)$	0	$\frac{1}{4}(1-{\Im }^2$)	cn ($\xi$)/[1 ± sn ($\xi$)]
6	$\frac{1}{4}$	0	$\frac{1}{2}(1+{\Im }^2)$	0	$\frac{1}{4}{(1-{\Im }^2)}^2$	sn ($\xi$)/[dn ($\xi$) ± cn ($\xi$)]
7	$\frac{1}{4}{\Im }^2$	0	$\frac{1}{2}({\Im }^2-2)$	0	$\frac{1}{4}$	ns ($\xi$) + ds ($\xi$)
8	$\frac{1}{4}{\Im }^2$	$-{\Im }^2$	$-(1-2{\Im }^2)$	$2(1-{\Im }^2)$	$-(1-{\Im }^2)$	(ℑ sn ($\xi$))/(ℑ sn ($\xi$) + dn ($\xi$) − 1)

and

Table 2. JEFS for $\Im \to 0$ and $\Im \to 1$.

		$\mathit{\Im }\to 0$	$\mathit{\Im }\to 1$			$\mathit{\Im }\to 0$	$\mathit{\Im }\to 1$
1	sn ($\xi$)	$\sin \left(\xi \right)$	$\tanh \left(\xi \right)$	7	dc ($\xi$)	$\sec \left(\xi \right)$	1
2	cn ($\xi$)	$\cos \left(\xi \right)$	sech ($\xi$)	8	nc ($\xi$)	$\sec \left(\xi \right)$	$\cosh \left(\xi \right)$
3	dn ($\xi$)	1	sech ($\xi$)	9	sc ($\xi$)	$\tan \left(\xi \right)$	$\sinh \left(\xi \right)$
4	cd ($\xi$)	$\cos \left(\xi \right)$	1	10	ns ($\xi$)	$\csc \left(\xi \right)$	$\coth \left(\xi \right)$
5	sd ($\xi$)	$\sin \left(\xi \right)$	$\sinh \left(\xi \right)$	11	ds ($\xi$)	$\csc \left(\xi \right)$	csch ($\xi$)
6	nd ($\xi$)	1	$\cosh \left(\xi \right)$	12	cs ($\xi$)	$\cot \left(\xi \right)$	csch ($\xi$)

, according to suitable choices of the free parameters existing in the obtained solutions [36]:

The absolute, the real part, the complex part of the solution given by (24) are depicted in Figure 3.

4. Conclusions

The fundamental purpose of this research is to use the unified method to acquire new non-autonomous implicit and explicit complex wave solutions for the Vc-GDNLSE in the context of optical fibers, particularly in the realm of optical networks. Solitons, exponential rational solutions, and elliptic (or periodic or hyperbolic) wave solutions are among the geometrical structures used to generate the outcomes of the polynomial type. 3D and 2D figures are presented to analyze the dynamical behavior for the obtained solutions by choosing suitable values for the free parameters and arbitrary functions included in these solutions. Our main strategy can minimize the number of computational predictions and is simple to use for a range of physical problems in the areas of optics and applied sciences. The results show that the suggested technique is successful, straightforward, and efficient.