Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

Abstract

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful ($G^{\prime }/G$, $1/G$)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters $\lambda$, $\tau$, $c$, $\mathcal{B}$, and $\mathsf{{\rm K}}$ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter $\lambda$ exhibits consistently high sensitivity across all scenarios, while parameters $\tau$ and $c$ demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

1. Introduction

Nonlinear equations (NLEs) are not just mathematical puzzles; instead, they are the hidden language of nature’s most captivating phenomena. Unlike the straightforward lines of linear equations, NLEs capture the intricate dances of heat flow, light waves, and more, revealing the true complexity of our world. These equations unlock the secrets of solitons, solitary waves that travel through optical fibers, and hold the key to unraveling countless scientific mysteries. From the scorching heat of a furnace to the shimmering ripples of light pulses, NLEs provide a powerful lens to understand and harness these phenomena. By deciphering their stories, mathematicians and engineers can build better fiber optic networks, design efficient heat transfer systems, and unlock the mysteries beyond the realm of light, shaping the future in remarkable ways. Furthermore, understanding and controlling soliton dynamics are crucial for various optical applications, encompassing optical communication, nonlinear optics, and ultrafast photonics. Optical solitons offer unique benefits, such as stable propagation over long distances and the preservation of pulse shape and integrity. As a result, accurate modeling of soliton dynamics is of paramount importance for advancing and optimizing optical devices and systems tailored for these specific applications [1,2,3,4]. Within the realm of optical fiber communication, the significance of the BME is notable, as it stands as a valuable tool for modeling diverse soliton solutions. These solitons represent solitary waves capable of traversing long distances without altering their form, a characteristic crucial for transmitting information without distortion [5,6]. Furthermore, their equations are integrable, indicating that they harbor an infinite number of conservation laws. This investigation delves into the interaction between parabolic and power law nonlinearities within the framework of the BME. Here is the form of the BME [5]:

$$i{\left({\mathcal{M}}^n\right)}_t-\left\{\left(\mathcal{B}\mathbb{C}{\left|\mathcal{M}\right|}^2\right)-\tau \right\}{\mathcal{M}}^n-\mathsf{{\rm K}}\left\{{\left({\mathcal{M}}^n\right)}_{xx}+{\left({\mathcal{M}}^n\right)}_{yy}\right\}=0; i^2=-1.$$

(1)

where $\mathcal{M}=\mathcal{M}(x,y,t)$ correspond to the complex function, and ${\left({\mathcal{M}}^n\right)}_t=\frac{\partial {\mathcal{M}}^n}{\partial t}$, ${\left({\mathcal{M}}^n\right)}_{xx}=\frac{{\partial }^2{\mathcal{M}}^n}{\partial x^2}$ and ${\left({\mathcal{M}}^n\right)}_{yy}=\frac{{\partial }^2{\mathcal{M}}^n}{\partial y^2}$. In Equation (1), there are three components: the first signifies the overall evolution, the second term encapsulates the function $\mathbb{C}$, representing the nonlinearity, and the third term encompasses the group velocity dispersion. Also, $\mathbb{C}$ is a real-valued algebraic function, and $n$ serves as a constraint that extends from the nonlinear Schrödinger equation (NLSE) to the BME. The variables $x$, $y$, and $t$ are independent spatial and temporal variables, while $\mathcal{B}$, $\mathsf{{\rm K}}$, and $\tau$ are real quantities.

Generally, $n$ is larger than or equal to 1, and once $n$ equals 1, Equation (1) shortens to the (2 + 1)-dimensional NLSE structure of the BME. This investigation centers on Equation (1), which shows evident nonlinearity, with the hypothesis that $n$ corresponds 1 is a basic NLSE form for optical systems, particularly optical fibers.

Numerous researchers have developed various methodologies to derive precise solutions for NLSEs, employing diverse approaches such as the $\frac{G^{\prime }}{G^{\prime }+G+A}$ method [7,8], the Sardar subequation method [9,10], the Riccati equation method [11,12], the Hirota bilinear method [13,14], the Lie group method [15,16], the $\left(G^{\prime }/G\right)$-expansion technique [17,18], the extended Jacobi elliptic function method [19,20], the functional variable technique [21,22], the homogeneous balance method [23,24], the Hirota bilinear formulation with N- soliton [25,26], the new auxiliary equation method [27,28], the tanh–function method [29,30,31], the tanh–coth method [32,33], the generalized Kudryshov method [34,35], the exp$\left(-\phi \left(\xi \right)\right)$- expansion method [36,37], the unified method [38,39], the multiple exp-function method [40,41] and so on.

Among various methods, the $\left(\frac{G^{\prime }}{G},\frac{1}{G}\right)$ -expansion method is a powerful analytical technique utilized in solving NLEs. It involves representing solutions as power series, where coefficients are figured out using two variables $\frac{G^{\prime }}{G}$ and $\frac{1}{G}$. These coefficients are obtained by substituting the series into the equation and matching coefficients of similar terms. This method is widely employed to reveal optical solutions effectively. Numerous researchers [42,43,44,45,46,47,48,49] have utilized this technique to reveal solutions for various NLEs. Throughout history, researchers have applied diverse mathematical models like the parabolic law, Kerr law, power law, and Kudryashov’s quintuple power law to probe optical soliton solutions within the framework of the BME equation, employing various solving techniques. While earlier investigations primarily centered on revealing bright and dark soliton solutions, our study takes a different approach, aiming to investigate a wider array of soliton patterns. This includes kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton patterns. As of present, there has been no investigation of BME employing this technique with Kerr law and power law nonlinearity. This investigation aims to use the above-mentioned methodology to derive soliton solutions for the provided nonlinear equations. In addition to our primary investigation, we embarked on a comprehensive first-order sensitivity analysis, colloquially termed local sensitivity analysis. This analysis aims to elucidate the nuanced impacts of minor perturbations in each input parameter on the resultant outcomes within a specific operational context. This analytical endeavor involves the meticulous computation of sensitivities about the output concerning each input parameter, employing numerical methodologies such as finite differences. By scrutinizing these sensitivities, we tried to unveil the intricate interplay between individual input parameters and their consequential effects on the overarching solutions. Such insights are pivotal in augmenting our understanding of the complex dynamics of the BME’s exact solutions. This paper is divided into several sections: (i) Methodology overview in Section 2. (ii) Discussion of parabolic and power law nonlinearity in Section 3. (iii) Application of the methodology to derive solutions from the equations in Section 4. (iv) Exploration of dynamic representations, showcasing soliton behaviors through 3D graphs, 2D graphs, and contours, along with sensitivity analysis of obtained solutions in Section 5. (v) Concluding remarks in Section 6. (vi) Finally, a list of relevant references.

2. A Short Overview of (${\mathit{G}}^{\mathbf{\prime }}\mathbf{/}\mathit{G}$, $\mathbf{1}\mathbf{/}\mathit{G}$)-Expansion Technique

Here, we provide an in-depth explanation of the basic steps required to apply the ($G^{\prime }/G$, $1/G$) technique, which was first suggested by [50], to the analysis of NLEs. To aid in this analytical process, a careful formulation of an auxiliary linear ordinary differential equation of the following form is required:

$$G^{″}\left(\phi \right)+\lambda G\left(\phi \right)=\eta ,$$

(2)

where ‘′’ denotes first derivatives concerning $\phi$, and the variables are:

$$\left.\begin{array}{c}\mathcal{g}=\frac{{\mathcal{G}}^{\prime }\left(\phi \right)}{G\left(\phi \right)}\\ F=\frac{1}{G\left(\phi \right)}\end{array}\right\}.$$

(3)

Equation (3) satisfies the subsequent correlations:

$$\left.\begin{array}{c}{\mathcal{g}}^{\prime }={-\mathcal{g}}^2+\eta F-\lambda \\ {\mathcal{F}}^{\prime }=-g\ast F\end{array}\right\}.$$

(4)

The result of Equation (2) mentioned earlier fluctuates depending on λ, which falls into three separate scenarios [43,51]:

Scenario I: For $\lambda >$ 0.

Equation (2) gives the following form of the general solution (GS):

$$G\left(\phi \right)=l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.,$$

(5)

which generates:

$${\mathcal{F}}^2=\left(\frac{{\mathcal{g}}^2-2\eta \mathcal{F}+\lambda }{{A_1\lambda }^2-{\eta }^2}\right)\lambda ,$$

(6)

where $A_1=l_1^2+l_2^2$ is the arbitrary constant.

Scenario II: For $\lambda <0$

Equation (2) yields the succeeding form of the GS:

$$G\left(\phi \right)=l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.,$$

(7)

subsequent in:

$${\mathcal{F}}^2=-\lambda \left(\frac{{\mathcal{g}}^2-2\eta \mathcal{F}+\lambda }{{A_2\lambda }^2+{\eta }^2}\right),$$

(8)

where $A_2=l_1^2-l_2^2$ is the arbitrary constant.

Scenario III: For $\lambda =0$

Equation (2) results in the following form of the GS:

$$G\left(\phi \right)=\frac{\eta {\phi }^2}{2}+l_1\phi +l_2,$$

(9)

which gives:

$${\mathcal{F}}^2=\left(\frac{{\mathcal{g}}^2-2\eta \mathcal{F}}{l_1^2-2\eta l_2}\right),$$

(10)

where $l_1$ and $l_2$ are the arbitrary constant.

Consider a general form of NLEs incorporating three independent variables ($x, y,$ and $t$) as follows:

$$L\left(\mathcal{M},{\mathcal{M}}_x,{\mathcal{M}}_{xx},{\mathcal{M}}_y,{\mathcal{M}}_{yy},{\mathcal{M}}_{xy},{\mathcal{M}}_t,{\mathcal{M}}_{tt},{\mathcal{M}}_{xt},{\mathcal{M}}_{yt}\right)=0,$$

(11)

where $L$ denotes a polynomial function reliant on the variables encompassed in $\mathcal{M}$ and ${\mathcal{M}}_x=\frac{\partial \mathcal{M}}{\partial x}$, ${\mathcal{M}}_y=\frac{\partial \mathcal{M}}{\partial y}$, ${\mathcal{M}}_t=\frac{\partial \mathcal{M}}{\partial t}$, ${\mathcal{M}}_{xx}=\frac{{\partial }^2\mathcal{M}}{\partial x^2}$, ${\mathcal{M}}_{yy}=\frac{{\partial }^2\mathcal{M}}{\partial y^2}$, ${\mathcal{M}}_{tt}=\frac{{\partial }^2\mathcal{M}}{\partial t^2}$, ${\mathcal{M}}_{xt}=\frac{\partial \mathcal{M}}{\partial x\partial t}$, ${\mathcal{M}}_{xy}=\frac{\partial \mathcal{M}}{\partial x\partial y}$ and so on.

To transform Equation (11), a new variable $\phi$ is introduced, governed by the following relation:

$$\mathcal{M}\left(x,y,t\right)=e^{i\mathsf{\theta }}\mathbb{Z}\left(\phi \right),$$

(12)

where $\theta =x+y+\nu t+{\sigma }_0$ and $\phi =cx+dy+\varpi t$.

Equation (12) portrays a wave characterized by a velocity of $\varpi$, phase described by $\theta$, and a fixed starting point defined by the phase constant ${\sigma }_0$. However, to completely grasp its behavior, three real values, $c$, $d,$ and $\nu$, remain unresolved and require determination.

Now, Equation (11) is converted by using the transformation written in Equation (12) to a new form as follows:

$$J\left(\mathbb{Z},{\mathbb{Z}}^{\prime },{\mathbb{Z}}^{″},{\mathbb{Z}}^{\prime \prime \prime },\dots \dots \dots \right)=0.$$

(13)

In this context, $J$ is the new polynomial encapsulating $\mathbb{Z}$ along with its ordinary derivatives.

The GS of Equation (13) is provided by the following equation, derived through the previously discussed method as follows:

$$\mathbb{Z}\left(\phi \right)=a_0+{\sum }_{i=1}^N{a}_i{\mathcal{g}}^i\left(\phi \right)+{\sum }_{i=1}^N{b}_i{\mathcal{g}}^{i-1}\left(\mathsf{\phi }\right)\mathcal{F}\left(\phi \right).$$

(14)

Here, $a_0$, $a_i$, and $b_i(i=1,2,3,\dots \dots \dots ,N)$ are the arbitrary constant fulfilling the requirement $a_N^2+b_N^2\ne 0$, and the parameter $N$ is defined as a positive balance number.

Exact Solution Computation Workflow:

First, applying the homogeneous balance principle to the transformed ODE provides the value of the balance number, then switching the value of $N$ into Equation (14) and integrating the modified equation into Equation (13), with the utilization of Equations (6), (8) and (10), the left-hand side of Equation (13) transforms a polynomial containing the terms $\mathcal{g}$ and $\mathcal{F}$. In this polynomial, the degree accompanying $\mathcal{F}$ does not exceed 1, while the degree of $\mathcal{g}$ ranges from $0$ to any integer. Establishing a system of algebraic equations involves equating the coefficients of terms with corresponding powers within the polynomial to $0$, incorporating parameters such as $a_i$, $b_i$, λ, $\eta$, and others. Employing Mathematica software 14 aids in solving these algebraic equations to determine the values of these parameters. Subsequently, these values are substituted into Equation (13), enabling the extraction of soliton solutions represented by trigonometric functions (as in Equation (5)), hyperbolic functions (as in Equation (7)), and rational functions (as in Equation (9)). Through the utilization of Equation (12), the exact solution of Equation (1) is derived, encompassing trigonometric, hyperbolic, and rational functions.

3. Implementation of Nonlinearities Laws

In this section, we analyzed Equation (1) using both the Ker and power law nonlinearity. The implementation of these laws on the BME is detailed in Section 3.1 and Section 3.2 below.

3.1. Ker Law

For an exploration of Equation (1) under Kerr law nonlinearity, specifically when $\mathbb{C}\left(\Theta \right)=\Theta$ [52,53], the resulting NLSE is formulated as follows:

$$i{\mathcal{M}}_t-\left\{\left(\mathcal{B}{\left|\mathcal{M}\right|}^2\right)-\tau \right\}\mathcal{M}-{\rm K}\left({\mathcal{M}}_{xx}+{\mathcal{M}}_{yy}\right)=0.$$

(15)

By inserting Equation (12) into Equation (15), we obtain the following equations from the imaginary and real components:

$$\left\{\varpi -2{\rm K}\left(c+d\right)\right\}{\mathbb{Z}}^{\prime }\left(\phi \right)=0,$$

(16)

and

$$\left(\tau -c+2{\rm K}\right)\mathbb{Z}\left(\phi \right)-\mathcal{B}{\left\{\mathbb{Z}\left(\phi \right)\right\}}^3-{\rm K}\left(c^2+d^2\right){\mathbb{Z}}^{″}\left(\phi \right)=0.$$

(17)

Given that $\mathbb{Z}\left(\phi \right)$ is non-zero and shows second-order derivatives, Equation (16) imposes the following restriction on the velocity:

$$\varpi =2\left(c+d\right){\rm K}.$$

(18)

Hence, Equation (17) signifies the NODE (nonlinear ordinary differential equation) form of Equation (15) with Kerr law nonlinearity, subject to the constraint outlined in Equation (18).

3.2. Power Law

In examining Equation (1) under power law nonlinearity, specifically when $\mathbb{C}\left(\Theta \right)={\Theta }^n$ [52,53], the resulting NLSE is formulated as follows:

$$i{\mathcal{M}}_t-\left\{\left(\mathcal{B}{\left|\mathcal{M}\right|}^{2n}\right)-\tau \right\}\mathcal{M}-{\rm K}\left\{{\mathcal{M}}_{xx}+{\mathcal{M}}_{yy}\right\}=0.$$

(19)

In this scenario, for stability, it is imperative that 0 < $n$ < 2. By inserting Equation (12) into Equation (19) and organizing the resulting expression, the following equations are derived:

$$\left\{\varpi -2{\rm K}\left(c+d\right)\right\}{\mathbb{Z}}^{\prime }\left(\phi \right)=0,$$

(20)

and

$$\left[-\mathcal{B}{\left\{\mathbb{Z}\left(\phi \right)\right\}}^{2n}+\tau -c+2{\rm K}\right]\mathbb{Z}\left(\phi \right)-\mathsf{{\rm K}}\left(c^2+d^2\right){\mathbb{Z}}^{″}\left(\phi \right)=0.$$

(21)

Likewise, Equation (21) offers the following restriction on the velocity:

$$\varpi =2\left(c+d\right){\rm K}.$$

(22)

Therefore, Equation (21) signifies the NODE form of Equation (19) with power law nonlinearity, subject to the constraint outlined in Equation (22).

4. Method’s Application with Nonlinearities Laws

In this segment, we employ the methodologies elucidated in Section 2 to derive the optical soliton solution for Equation (1).

4.1. Application for Kerr Law

Through the application of the balancing principle in Equation (17) provides $N=1$. Therefore, Equation (14) transforms into the following form:

$$\mathbb{Z}\left(\phi \right)=a_0+a_1\mathcal{g}\left(\phi \right)+b_1\mathcal{F}\left(\phi \right).$$

(23)

In this equation, the coefficients $a_0$, $a_1$ and $b_1$ are constants that need to be decided.

Scenario I. $\lambda >0$ (For trigonometric)

To obtain the solutions for Equation (17), we use Equations (4) and (6) and replace Equation (17) with Equation (23). With this technique, a polynomial with $\mathcal{g}$ and $\mathcal{F}$ is created from the left side of Equation (17). This polynomial’s coefficients, $a_0$, $a_1$ and $b_1$, are involved in a system of algebraic equations when each coefficient is set to zero. Solving this system of algebraic equations yields the follows arbitrary constants:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\\ { b}_1=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{{\lambda }^2{\mathrm{A}}_1-\eta }^2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{\lambda }}\end{array}\right\},$$

where $\tau =\frac{1}{2}(2c-4{\rm K}+c^2{\rm K}\lambda +d^2{\rm K}\lambda )$ and ${\rm K}<0.$

Now, using these calculated values in Equation (14), we obtain the following solution of Equation (17):

$$\begin{gathered} \mathbb{Z}\left(\phi \right)=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\sqrt{\lambda }\frac{l_{1}cos\left(\phi \sqrt{\lambda }\right)-l_{2}sin\left(\phi \sqrt{\lambda }\right)}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{{\lambda }^2{A}_1-\eta }^2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{\lambda }}\frac{1}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}, \end{gathered}$$

(24)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$.

Reverting to its first form with the aid of Equation (12), we obtain the following form:

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\theta }\{\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\sqrt{\lambda }\frac{l_{1}cos\left(\phi \sqrt{\lambda }\right)-l_{2}sin\left(\phi \sqrt{\lambda }\right)}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{{\lambda }^2{A}_1-\eta }^2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{\lambda }}\frac{1}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\}, \end{gathered}$$

(25)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

Scenario II. $\lambda <0$ (For hyperbolic)

To derive the hyperbolic solution, apply the same procedure, resulting in:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\\ { b}_1=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{\eta }^2+{\lambda }^2{\mathrm{A}}_2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{-\lambda }}\end{array}\right\},$$

where $\tau =\frac{1}{2}(2c-4{\rm K}+c^2{\rm K}\lambda +d^2{\rm K}\lambda )$ and ${\rm K}<0.$

Now, using these calculated values in Equation (14), the following solution of Equation (17):

$$\begin{gathered} \mathbb{Z}\left(\phi \right)=\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\sqrt{-\lambda }\frac{l_{1}cosh\left(\phi \sqrt{-\lambda }\right)+l_{2}sinh\left(\phi \sqrt{-\lambda }\right)}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{\eta }^2+{\lambda }^2{A}_2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{-\lambda }}\frac{1}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}, \end{gathered}$$

(26)

where $\mathsf{\phi }=cx+dy+2\left(c+d\right){\rm K}t$.

With the help of Equation (12), obtain solution of main PDEs:

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}}{\sqrt{2}\sqrt{\mathcal{B}}}\sqrt{-\lambda }\frac{l_{1}cosh\left(\phi \sqrt{-\lambda }\right)+l_{2}sinh\left(\phi \sqrt{-\lambda }\right)}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{-\left(c^2+d^2\right){\rm K}}\sqrt{{\eta }^2+{\lambda }^2{A}_2}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{-\lambda }}\frac{1}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\right\} \end{gathered}$$

(27)

where $\mathsf{\phi }=cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

When $l_1$ is confirmed to be non-zero and both $\eta$ and $l_2$ are set to zero, Equation (27) takes on the form of the following soliton solution:

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\mathsf{\theta }}\left\{\pm \frac{\sqrt{-\left(c^2+d^2\right)\mathsf{{\rm K}}}}{\sqrt{2}\sqrt{\mathcal{B}}}\sqrt{-\lambda }coth\left(\mathsf{\phi }\sqrt{-\lambda }\right)\pm \\ \frac{\sqrt{-\left(c^2+d^2\right)\mathsf{{\rm K}}}}{\sqrt{2}\sqrt{\mathcal{B}}\sqrt{-\lambda }}\lambda cosech\left(\mathsf{\phi }\sqrt{-\lambda }\right)\right\}, \end{gathered}$$

(28)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

Scenario III. For $\lambda =0$ (For rational)

To achieve the rational solution, follow the same procedure outlined in scenario I, resulting in:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{\mathcal{B}}}\\ { b}_1=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{{\mathrm{l}}_1^2-2l_2\eta }}{2\sqrt{\mathcal{B}}}\end{array}\right\},$$

where ${\rm K}=\frac{c-\tau }{2}$.

Now, incorporating these computed values into Equation (14), we derive the following solution for Equation (17):

$$\mathbb{Z}\left(\mathsf{\phi }\right)=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{\mathcal{B}}}\frac{2\eta \mathsf{\phi }+l_1}{\frac{\eta }{2}{\mathsf{\phi }}^2+l_1\mathsf{\phi }+l_2}\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{{\mathrm{l}}_1^2-2l_2\eta }}{2\sqrt{\mathcal{B}}}\frac{1}{\frac{\eta }{2}{\mathsf{\phi }}^2+l_1\mathsf{\phi }+l_2},$$

(29)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$.

Inserting the transformation variables defined in Equation (12) into the above equation results in the following generalized solution for Equation (1):

$$\mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{\mathcal{B}}}\frac{2\eta \phi +l_1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2}\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{l_1^2-2l_2\eta }}{2\sqrt{\mathcal{B}}}\frac{1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2}\right\},$$

(30)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

When $l_1$ is confirmed to be non-zero and both $\eta$ and $l_2$ are set to zero, Equation (30) takes on the form of the following soliton solution:

$$\mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{\mathcal{B}}}\frac{1}{\phi }\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{\mathcal{B}}}\frac{1}{\phi }\right\},$$

(31)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

4.2. Application for Power Law

Operating the balancing principle in Equation (21), we deduce that $N=1/n$. Consequently, the following relation can be defined:

$$\mathbb{Z}\left(\mathsf{\phi }\right)={\mathsf{{\rm Y}}\left(\mathsf{\phi }\right)}^{\frac{1}{n}}.$$

(32)

Thus, employing the transformation, Equation (38) undergoes the following form:

$$\begin{gathered} \left(\tau -c+2{\rm K}\right)n^2{\left\{{\rm Y}\left(\phi \right)\right\}}^2-\mathcal{B}{n}^2{\left\{{\rm Y}\left(\phi \right)\right\}}^4+{\rm K}\left(c^2+d^2\right)\left(n-1\right){\left\{{{\rm Y}}^{\prime }\left(\phi \right)\right\}}^2-n{\rm K}\left(c^2+ \\ {d}^2\right){\rm Y}\left(\phi \right){{\rm Y}}^{″}\left(\phi \right)=0. \end{gathered}$$

(33)

Applying the balancing principle once more and figuring out $N=1$, the solution adopts the following structure:

$${\rm Y}\left(\phi \right)=a_0+a_1\mathcal{g}\left(\phi \right)+b_1\mathcal{F}\left(\phi \right).$$

(34)

Now, employing a similar procedure as described in Section 4.1 for all scenarios, we derive the following solutions for Equation (33), then Equation (21), and finally Equation (1):

Scenario I. $\lambda >0$ (For trigonometric)

The coefficients obtained from this case are as follows:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}\sqrt{\lambda }}\\ { b}_1=\pm \frac{\sqrt{c-\tau -2{\rm K}}\sqrt{{\lambda }^2{A}_1{-\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\end{array}\right\},$$

where $d=\pm \frac{\sqrt{2c-2\tau -4{\rm K}+c^2{\rm K}\lambda }\sqrt{{\lambda }^2{A}_1{-\eta }^2}}{\sqrt{\lambda }\sqrt{{\rm K}{\eta }^2-{\rm K}{\lambda }^2{A}_1}}$ and $n=1.$

Now, utilizing these computed values in Equation (34), we derive the following solution for Equation (33):

$$\begin{gathered} {\rm Y}\left(\phi \right)=\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}\sqrt{\lambda }}\sqrt{\lambda }\frac{l_{1}cos\left(\phi \sqrt{\lambda }\right)-l_{2}sin\left(\phi \sqrt{\lambda }\right)}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{c-\tau -2{\rm K}}\sqrt{{\lambda }^2{A}_1{-\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\frac{1}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}, \end{gathered}$$

(35)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$.

Returning to its original form with the aid of Equation (12), we obtain the following equation:

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}}\frac{l_{1}cos\left(\phi \sqrt{\lambda }\right)-l_{2}sin\left(\phi \sqrt{\lambda }\right)}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{c-\tau -2{\rm K}}\sqrt{{\lambda }^2{A}_1{-\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\frac{1}{l_{1}sin\left(\phi \sqrt{\lambda }\right)+l_{2}cos\left(\phi \sqrt{\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\right\}, \end{gathered}$$

(36)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

When $l_2$ is confirmed to be non-zero and both $\eta$ and $l_1$ are set to zero, Equation (36) takes on the form of the following soliton solution:

$$\mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}}tan\left(\phi \sqrt{\lambda }\right)\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}}sec\left(\phi \sqrt{\lambda }\right)\right\},$$

(37)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

When $l_1$ is confirmed to be non-zero and both $\eta$ and $l_2$ are set to zero, Equation (36) takes on the form of the following soliton solution:

$$\mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}}cot\left(\phi \sqrt{\lambda }\right)\pm \frac{\sqrt{c-\tau -2{\rm K}}}{\sqrt{\mathcal{B}}}cosec\left(\phi \sqrt{\lambda }\right)\right\},$$

(38)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

Scenario II. When $\lambda <0$ (For hyperbolic)

The obtained coefficients for this case are as follows:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{\tau +2{\rm K}-c}}{\sqrt{\mathcal{B}}\sqrt{-\lambda }}\\ { b}_1=\pm \frac{\sqrt{\tau +2{\rm K}-c}\sqrt{{\lambda }^2{\mathrm{A}}_2{+\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\end{array}\right\},$$

where $d=\pm \frac{\sqrt{2c-2\tau -4{\rm K}+c^2{\rm K}\lambda }\sqrt{{\lambda }^2{A}_1{+\eta }^2}}{\sqrt{-\lambda }\sqrt{{\rm K}{\eta }^2+{\rm K}{\lambda }^2{A}_2}}$ and $n=1.$

Now, incorporating these calculated values into Equation (34), we derive the following solution for Equation (33):

$$\begin{gathered} {\rm Y}\left(\phi \right)=\pm \frac{\sqrt{\tau +2{\rm K}-c}}{\sqrt{\mathcal{B}}\sqrt{-\lambda }}\sqrt{-\lambda }\frac{l_{1}cosh\left(\phi \sqrt{-\lambda }\right)+l_{2}sinh\left(\phi \sqrt{-\lambda }\right)}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{\tau +2{\rm K}-c}\sqrt{{\lambda }^2{A}_2{+\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\frac{1}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}, \end{gathered}$$

(39)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$.

Returning to its original form with the aid of Equation (12), we obtain the following equation:

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\theta }\left\{\pm \frac{\sqrt{\tau +2{\rm K}-c}}{\sqrt{\mathcal{B}}}\frac{l_{1}cosh\left(\phi \sqrt{-\lambda }\right)+l_{2}sinh\left(\phi \sqrt{-\lambda }\right)}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\pm \\ \frac{\sqrt{\tau +2{\rm K}-c}\sqrt{{\lambda }^2{A}_2{+\eta }^2}}{\sqrt{\mathcal{B}}\lambda }\frac{1}{l_{1}sinh\left(\phi \sqrt{-\lambda }\right)+l_{2}cosh\left(\phi \sqrt{-\lambda }\right)+\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\right\}, \end{gathered}$$

(40)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

When $l_1$ is confirmed to be non-zero and both $\eta$ and $l_2$ are set to zero, Equation (40) takes on the form of the following soliton solution:

$$\mathcal{M}\left(x,y,t\right)=e^{i\mathsf{\theta }}\left\{\pm \frac{\sqrt{\tau +2\mathsf{{\rm K}}-c}}{\sqrt{\mathcal{B}}}coth\left(\mathsf{\phi }\sqrt{-\lambda }\right)\pm \frac{\sqrt{\tau +2\mathsf{{\rm K}}-c}}{\sqrt{\mathcal{B}}}cosech\left(\mathsf{\phi }\sqrt{-\lambda }\right)\right\},$$

(41)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

Scenario III. If $\lambda =0$ (For rational)

The obtained coefficients for this case are as follows:

$$\left.\begin{array}{c}{a}_0=0\\ a_1=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{1+n}}{2\sqrt{2}n\sqrt{\mathcal{B}}}\\ { b}_1=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{{\mathrm{l}}_1^2-2l_2\eta }}{2\sqrt{2}n\sqrt{\mathcal{B}}}\end{array}\right\},$$

where ${\rm K}=\frac{c-\tau }{2}$.

Now, using these calculated values in Equation (34), we obtain the following solution for Equation (33):

$${\rm Y}\left(\phi \right)=\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{1+n}}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{2\eta \phi +l_1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2}\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{l_1^2-2l_2\eta }}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2},$$

(42)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$.

Plugging in the transformation variables outlined in Equation (12) into the given equation produces the following comprehensive solution for Equation (1):

$$\begin{gathered} \mathcal{M}\left(x,y,t\right)=e^{i\theta }{\left\{\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{1+n}}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{2\eta \phi +l_1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2}\pm \\ \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{l_1^2-2l_2\eta }}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{1}{\frac{\eta }{2}{\phi }^2+l_1\phi +l_2}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}, \end{gathered}$$

(43)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

Setting both $\eta$ and $l_2$ to zero, while ensuring $l_1$ is non-zero, Equation (30) simplifies to yield the soliton solution in the following manner:

$$\mathcal{M}\left(x,y,t\right)=e^{i\theta }{\left\{\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}\sqrt{1+n}}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{1}{\phi }\pm \frac{\sqrt{c^2+d^2}\sqrt{\tau -c}}{2\sqrt{2}n\sqrt{\mathcal{B}}}\frac{1}{\phi }\right\},}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$$

(44)

where $\phi =cx+dy+2\left(c+d\right){\rm K}t$ and $\theta =x+y+\nu t+{\sigma }_0$.

5. Graphs and Meanings

Harnessing the power of Mathematica, an advanced mathematical computing tool, we explored the intricate graphical patterns shown by the BME with Kerr law nonlinearity and BME with Power law nonlinearity. Our presentation encompasses a variety of visual representations, including 3D renderings, 2D graphical displays, and contour plots. These visualizations cover a broad range of parameter values for each relevant variable, allowing us to gain a thorough insight into the graphical behavior of these nonlinear equations across diverse parameter space.

To ensure clarity and conciseness, we opted for three representative solution sets for each case (BME with Kerr law nonlinearity and BME with power law nonlinearity) from our comprehensive results for visual representation, standardizing only three graphs, including 3D renderings, 2D graphical displays, and contour plots, for each nonlinearity. The specific constants linked to each graph are detailed in the corresponding figure captions. In 2D graphs, we consolidated multiple solutions within a single figure by varying the parameter t.

5.1. Visualization of the BME with Kerr Law Nonlinearity

The depiction in Figure 1, originating from Equation (25), vividly captures the essence of periodic bright solitons. These solitons, known for their periodic behavior and stable propagation, hold immense utility across various domains such as optical communications, nonlinear optics, and fiber optics. Their unique ability to convey information over extended distances without distortion is highly advantageous. Moreover, they serve as invaluable tools for studying wave dynamics and nonlinear phenomena, shedding light on the intricate behavior of light pulses within optical fibers and other waveguides.

The image portrayed in Figure 2, stemming from Equation (28), vividly illustrates the essence of kink-type solitons with singular characteristics. These solitons are recognized by their localized wave packets with a kink-shaped profile, showcasing distinctive behavior during optical fiber propagation. Renowned for their singular attributes, they hold significance across diverse optical fiber applications, spanning signal processing, nonlinear optics, and optical communications. Their exceptional features make them invaluable for enhancing the stability and efficiency of optical signal manipulation and transmission within fiber optic networks.

Figure 3, originating from Equation (30), showcases dark-type solitons. These solitons, characterized by localized waveforms with reduced intensities, are essential for various optical communication tasks such as signal amplification, pulse compression, and signal regeneration within fiber optic networks. Their unique ability to keep waveform integrity during propagation through optical fibers ensures reliable and efficient data transmission over long distances in optical communication systems. Geometrically, a dark soliton is a depressed section within a wave, where the amplitude falls below the neighboring crests, creating a noticeable trough. This imagery mirrors a dim, less intense area amid the brighter wave peaks, showcasing the soliton’s unique trait of maintaining its shape and conserving energy while moving through the medium.

5.2. Sensitivity Analysis

Sensitivity analysis involves scrutinizing how variations in input parameters impact the output of a model or system within a localized region of the parameter space. In recent studies, researchers [54,55] have utilized this analysis to explore how adjustments in input parameters influence the output of NLEs. In this section, we conduct a local sensitivity analysis of our derived solution for each scenario (for $\lambda =0, \lambda <0,$ and $\lambda >0$). Figure 4a illustrates the sensitivity of each parameter in Equation (25) for Scenario I. We note that parameters $\left(\lambda , c, d, l_1, \mathsf{{\rm K}}\right)$ show negative sensitivity, whereas $\lambda$ demonstrates notably high negative sensitivity. On the other hand, the remaining parameters $\left(\mathcal{B},\eta , l_2\right)$ exhibit positive sensitivity, although not as pronounced as $\lambda$. Additionally, it is apparent that $l_2$ exhibits almost negligible sensitivity compared to others. For Scenario II ($\lambda <0$), the results remain largely consistent, with one notable difference: the parameter $\mathsf{{\rm K}}$ shows slightly higher sensitivity compared to $\lambda$, as depicted in Figure 4b. In the case of a rational solution ($\lambda =0$), the most sensitive parameters are $c$ and $\tau$, but they exhibit opposite directions of sensitivity, whereas other parameters are nearly negligible, as illustrated in Figure 4c.

5.3. The Graphical Representation of the BME with Power Law Nonlinearity

Figure 5, originating from Equation (38), depicts a singular soliton type. This variant of solitons within optical fibers enables the transmission of information over extensive distances with minimal distortion, representing notable advancements in telecommunications and optical signal processing. Renowned for their ability to preserve shape and velocity during propagation, these solitary waves play a critical role in elevating data transmission rates and mitigating signal degradation in fiber optic communication systems.

The anti-kink soliton solution, derived from Equation (40) and depicted in Figure 6, is vital in optical fibers. It effectively counters kinks or distortions, preserving signal integrity and ensuring reliable long-distance communication. This is crucial in high-speed networks and fiber optic systems where signal degradation is detrimental. Additionally, anti-kink solitons enhance overall efficiency and stability, supporting the advancement of optical fiber-based technologies and high-capacity data transmission networks.

The solution derived from Equation (43) and illustrated in Figure 7 is of the hyperbolic type. Within optical fibers, this type of solution ensures stable signal transmission by preserving shape and velocity, minimizing distortion across long distances. Offering high energy and bandwidth, hyperbolic solutions facilitate efficient signal amplification and support high-capacity data transmission. Their robust characteristics enable extended transmission distances and improved signal quality. Moreover, hyperbolic solitons enable nonlinear effects such as soliton self-frequency shifting and supercontinuum generation, broadening their applications in nonlinear optics and photonics. Overall, they significantly elevate the performance and capabilities of optical fiber communication networks.

5.4. Sensitivity Analysis

Similarly, we performed a sensitivity analysis on our derived solution for each scenario of $\lambda =0, \lambda <0,$ and $\lambda >0$ for the power law nonlinearity of BME. In Figure 8a, we depict the sensitivity of each parameter in Equation (36) for Scenario I. Notably, parameters $\left(\lambda , c, d, \eta , \tau , \mathsf{{\rm K}}\right)$ display negative sensitivity, with λ showing particularly high negative sensitivity. Conversely, the remaining parameters $\left(\mathcal{B}, l_1, l_2\right)$ exhibit positive sensitivity, albeit not as pronounced as λ. Moreover, $l_1$ demonstrates almost negligible sensitivity compared to others. For Scenario II $(\lambda <0),$ the results remain largely consistent, with a notable difference: the parameter $c, d$ shows positive sensitivity, as depicted in Figure 8b. In the case of a rational solution $(\lambda =0),$ the most sensitive parameters are c and τ, but they exhibit opposite directions of sensitivity, whereas other parameters are nearly negligible, as illustrated in Figure 8c, which is similar to the Kerr law nonlinearity of BME sensitivity.

6. Conclusions

The ($G^{\prime }/G$, $1/G$) method was applied, and several precise optical soliton solutions were obtained for the Kerr and power law nonlinearities of BME. We used an analytical approach to generate several types of nonlinear wave patterns inside these nonlinearities, and different exact solutions in rational, hyperbolic, and trigonometric forms were revealed. Additionally, a sensitivity analysis was conducted to identify parameters that were more sensitive in both the negative and positive senses. The analysis underscores the profound influence of parameters $\lambda$, $\tau$, $c$, $\mathcal{B}$, and $\mathsf{{\rm K}}$ on the precise solutions derived across both Kerr law and power law nonlinearities within the context of BME. Notably, parameter $\lambda$ emerges as particularly critical across all scenarios, exhibiting heightened sensitivity. Moreover, parameters $\tau$ and $c$ demonstrate heightened sensitivity specifically within scenario III. These findings underscore the pivotal role of these parameters in shaping the outcomes of the studied phenomena, offering valuable insights into the nuanced interplay of resultant solutions. Validation through Mathematica software 14 ensured precision, while dynamic visual representations illustrated soliton solutions exhibiting different patterns, including bright solitons, dark solitons, singular solitons, king solitons, anti-kink, and parabolic types. These soliton solutions offer advantages such as energy localization, propagation stability, predictable interactions, precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, and suitability for optical signal processing in optical physics. The outcomes of this study are novel and hold promise for various forms of BME and related research in optical physics. Moreover, the demonstrated effectiveness and robustness of the applied method in obtaining diverse solitons within different forms of BME further underscore its significance.