Chaotic Behaviour, Sensitivity Assessment, and New Analytical Investigation to Find Novel Optical Soliton Solutions of M-Fractional Kuralay-II Equation

Abstract

The implementation of chaotic behavior and a sensitivity assessment of the newly developed M-fractional Kuralay-II equation are the foremost objectives of the present study. This equation has significant possibilities in control systems, electrical circuits, seismic wave propagation, economic dynamics, groundwater flow, image and signal denoising, complex biological systems, optical fibers, plasma physics, population dynamics, and modern technology. These applications demonstrate the versatility and advantageousness of the stated model for complex systems in various scientific and engineering disciplines. One more essential objective of the present research is to find closed-form wave solutions of the assumed equation based on the $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach. The results achieved are in exponential, rational, and trigonometric function forms. Our findings are more novel and also have an exclusive feature in comparison with the existing results. These discoveries substantially expand our understanding of nonlinear wave dynamics in various physical contexts in industry. By simply selecting suitable values of the parameters, three-dimensional (3D), contour, and two-dimensional (2D) illustrations are produced displaying the diagrammatic propagation of the constructed wave solutions that yield the singular periodic, anti-kink, kink, and singular kink-shape solitons. Future improvements to the model may also benefit from what has been obtained as well. The various assortments of solutions are provided by the described procedure. Finally, the framework proposed in this investigation addresses additional fractional nonlinear partial differential equations in mathematical physics and engineering with excellent reliability, quality of effectiveness, and ease of application.

1. Introduction

As a result of breakthroughs in modern science and technology, academics and researchers are beginning to understand natural occurrences better by utilizing partial differential equations (PDEs). Nonlinear partial differential equations (NPDEs) are a class of PDEs that have applications in plasma physics, ocean engineering, fluid dynamics, hydrodynamics, optical fibers, general relativity, complex acoustics, quantum mechanics, biology, geochemistry, finance, and several other natural contexts. Solitons have a wide range of applications across various fields due to their unique properties of shape preservation and constant speed. Research and analysis of soliton waves represent a domain in which NPDEs have been frequently employed. Several nonlinear real-world models have been implemented by scholars to interpret and forecast the characteristics of soliton waves [1,2,3].

In 1974, the first-ever international conference on fractional calculus was held. A fractional derivative generalizes a classical integer-order derivative to non-integer orders. Prominent mathematicians including Lagrange, Abel, Euler, Liouville, Riemann, Caputo, and numerous others contributed to this development. Different types of fractional derivatives are defined by Riemann–Liouville, Caputo, Hadamard, Caputo–Hadamard, Riesz, Atangana, Khalil, Sousa, and Oliveira. The unique characteristics of different definitions contribute to fractional calculus’s applicability in practical as well as theoretical contexts. It provides a general tool for modeling complex systems with non-integer differentiation orders, while M-fractional derivatives offer further customization to adapt these models to specific needs or behaviors. Both are vital in extending the applicability of traditional calculus to more complex, real-world problems. The application of fractional derivatives extends beyond traditional calculus in various fields such as physics, engineering, control systems, signal processing, biological systems, finance, electromagnetics, heat and mass transfer, and quantum mechanics. A nonlinear fractional partial differential equation (NFPDE) is a type of NPDE that incorporates fractional derivatives into a nonlinear equation. NFPDEs extend the modeling capability of traditional NPDEs, making them suitable for a wide range of real-world phenomena that exhibit complex, non-local, and memory-dependent behaviors. The branch of nonlinear sciences has paid close attention to NFPDEs over the past decade because of the magnificent variety of possibilities associated with them. In most natural disciplines, the closed-form solutions of NFPDEs have become vital to understanding complex nonlinear wave procedures. Due to their complexity, NFPDEs are an active area of research, and new techniques are constantly being developed to understand and solve them [4,5,6,7,8].

Several different approaches to analyze NFPDEs provide closed-form wave solutions, like the exp-function method [9], the variational iteration scheme [10], the r + mEDAM method [11], the bilinear neural network method [12], the bilinear residual network method [13], the sub-equation method [14], the method of Painleve analysis [15,16], the Lie symmetric analysis [17], the Sardar subequation scheme [18], the rational sine-Gordon expansion method [19], the advanced exp(-$\varphi$($\xi$))-expansion scheme [20], the ${\varphi }^6$ model expansion method [21], the Laplace homotopy analysis method [22], the improved tanh$({\displaystyle \frac{\phi (\xi )}{2}})$-expansion method [23], the simplified Hirota method [24], the modified extended mapping method [25], the (${\displaystyle \frac{G^{\prime }}{G^2}}$)-expansion method [26], the improved auxiliary equation method [27], the F-expansion method [28], the extended hyperbolic function method [29], the generalized exponential rational function method [30], the enhanced algebraic method [31], and many others. Some other interesting wave solutions have been explored in [32,33,34]. The $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion strategy was recently introduced and has become a highly prevalent mathematical technique for figuring out NPDEs. It has proven very constructive within the field of nonlinear mathematical analysis. Several years ago, Hong et al. [35] successfully applied the $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach to generate new exact solutions to the two forms of the Schrödinger equations. This approach has been confirmed by scholars to be advantageous. In an investigation, Mia et al. [36] employed this technique and produced excellent findings. This approach has been confirmed to be practical and effective, and it has been employed by multiple investigators to study NPDEs [37,38]. In a recently published investigation, Borhan et al. [39] adopted this technique to discover the new optical soliton structures of the (3 + 1)-dimensional Kadomtsev–Petviashvili equation and the (3 + 1)-dimensional Jimbo–Miwa equation.

The M-fractional Kuralay-II equation (KIIE) takes as its starting point efforts to generalize NPDEs by incorporating the M-fractional derivative, which better reveals nonlocal memory effects in complex physical systems. It assists as a platform to discover soliton solutions and memory-influenced dynamics exploiting modern analytical and computational tools. This model is an advanced mathematical equation that could be developed further to address a wide range of nonlinear fractional differential equations. Its ability to handle complex situations has made it a promising model for future research in mathematics, physics, engineering, and beyond. This equation is used in modeling complex systems and real-world systems with greater precision and flexibility. Professor Kuralay Yeskermessova is the founder of this model. The model is integrable. It has two types: the Kuralay-IIA (KIIA) equation and the Kuralay-IIB (KIIB) equation. It is utilized for modeling and analyzing systems with complex, non-local, and memory-dependent dynamics across a wide range of disciplines, including viscoelastic materials, electrical circuits, complex biological systems, pharmacokinetics, economic dynamics, control systems, plasma physics, nonlinear optics, image and signal denoising, population dynamics, and ferromagnetic materials. Various types of research on this concept have been documented in the literature. Applying the Hirota bilinear method, Sagidullayeva et al. [40] established the soliton solutions of the mentioned model. Zafar et al. [41] implemented three methods to identify the closed-form solutions of the KIIE. Some other scholars have explored the wave solutions of the KIIE through several schemes [42,43,44]. Recently, in [45], the closed-form wave solutions of the stated model were discovered exercising the Jacobi elliptic function expansion process.

In chaotic analysis, even minor modifications in input may generate wildly distinctive outcomes [46,47,48]. Because of its high sensitivity to initial conditions, this can produce radically varied outcomes progressively with slight adjustments to its initial parameters. In addition to being sensitive to disturbances, these kinds of structures frequently demonstrate patterns that are both complicated and unstructured. Several natural and manmade systems, such as climate conditions, turbulent fluid flows, demographics, and financial markets, are examples of areas where it might be detected. The sensitivity assessment analyzes how input unpredictability affects the outcomes [8,39,49]. The Runge–Kutta method has been employed to characterize the nonlinear system’s sensitivity. We utilize it in finance, engineering, economics, and the scientific community to evaluate decisions or deliberately model resilience under multiple circumstances.

From the above review, we observe that until now no one has inspected the chaotic analysis, the sensitivity assessment, and the closed-form wave solutions of the stated model utilizing the M-fractional derivative and the anticipated technique. The current research is inspired mostly by the uniqueness of our constructed solutions, which have unique features distinct from the existing outcomes. Researching novel closed-form wave solutions to the KIIE (KIIA and KIIB equations) employing the M-fractional derivative and the previously claimed expansion method is the key objective of the present investigation. Since the truncated M-fractional derivative, alongside the employed methodology, is novel for the model, our produced solutions vary from the current outcomes in that they are more recent, appealing, and inclusive. In $3D$, contour, and $2D$ formats, we illustrate such solutions to demonstrate the dynamics of wave transmission. The obtained outcomes are highly valuable for the model’s additional inquiry. Moreover, we see that the utilized approach is relatively simple, accessible to implement, and practical for analyzing the closed-form solutions of other NFPDEs that frequently occur in applied mathematics, modern physics, and contemporary engineering technology.

The current paper has various sections: The introduction has been given in Section 1. The notion of a truncated M-fractional derivative and its properties are described in Section 2. Section 3 is arranged with a fundamental discussion of the expansion procedure. Specification of the model is furnished in Section 4. Also, an application of the expansion technique is provided in Section 5. Moreover, diverse analyses of the model equation are given in Section 6. Section 7 is designed for graphical explanation and discussion. Finally, a conclusion is offered in Section 8.

2. Notion of Truncated M-Fractional Derivative and Its Properties

Fractional derivatives extend the concept of ordinary differentiation to non-integer orders. This concept is highly useful in various fields such as physics, engineering, biology, finance, control systems, signal processing, electromagnetics, heat and mass transfer processes, and quantum mechanics. Riemann–Liouville, Caputo, Hadamard, Caputo–Hadamard, Kober, Riesz, Katugampola, and Atangana–Baleanu have already established several definitions along with properties of fractional derivatives. A relatively recent proposal that focuses on the classical features of calculus was made by Khalil et al. [50] and has been referred to as the conformable fractional derivative. The M-truncated fractional derivative is a recent concept introduced to generalize the definitions of a conformable fractional derivative [51]. The M-truncated fractional derivative provides a practical modification to the classical fractional derivatives by making them more suitable for real-world applications that require computational efficiency. The M-derivative is more closely related practical, real-world contexts such as viscoelastic behavior, relaxation, diffusion, biological systems with memory, electrical circuits with fractional behavior, anomalous transport, and heat conduction [41,43,52].

Here, we introduce the truncated Mittag–Leffler function with a single parameter [51],

$${\mathfrak{E}}_{\chi }(z)=\sum _{k=0}^l{\displaystyle \frac{z^k}{\Gamma (\chi k+1)}},$$

(1)

in which $\chi >0$ and $z\in \mathbb{C}$. Suppose $v(\vartheta ):[0,\infty )\to \mathbb{R}$; then the M-truncated fractional derivative might be defined for v of order $\gamma$ [53]:

$${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }v(\vartheta )=\underset{\sigma \to 0}{lim}{\displaystyle \frac{v(\vartheta {\mathfrak{E}}_{\chi }(\sigma {\vartheta }^{1-\gamma }))-v(\vartheta )}{\sigma }},$$

(2)

wherein $0<\gamma \le 1,\chi >0$.

When $v_1$ and $v_2$ are $\gamma$-differentiable at any point $\vartheta >0$ with $0<\gamma \le 1,\chi >0$, $a_1,a_2\in \mathbb{R}$, we have the following properties of the M-truncated fractional derivative [53]:

• ${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }(a_1{v}_1(\vartheta )+a_2{v}_2(\vartheta ))=a_1{\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_1(\vartheta )+a_2{\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_2(\vartheta )$.
• ${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }(v_1(\vartheta ).v_2(\vartheta ))=v_1(\vartheta ){\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_2(\vartheta )+v_2(\vartheta ){\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_1(\vartheta )$.
• ${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }({\displaystyle \frac{v_1(\vartheta )}{v_2(\vartheta )}})={\displaystyle \frac{v_2(\vartheta ){\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_1(\vartheta )-v_1(\vartheta ){\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_1(\vartheta )}{{(v_2(\vartheta ))}^2}}$.
• ${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }(A_1)=0$—in this case $A_1$ is a constant.
• ${\mathbb{D}}_{M,\vartheta }^{\gamma ,\chi }{v}_1(\vartheta )={\displaystyle \frac{{\vartheta }^{1-\gamma }}{\Gamma (1+\chi )}}{\displaystyle \frac{\mathrm{d}{v}_1(\vartheta )}{\mathrm{d}\vartheta }}$.

3. Fundamental Discussion of the Expansion Procedure

In this section of this research paper, we offer a detailed overview of the technique that we considered using while looking for promising findings for NFPDEs. Firstly, we pick an NFPDE as follows:

$$\mathcal{F}(v,{\mathbb{D}}_{M,x}^{\gamma ,\chi }v,{\mathbb{D}}_{M,t}^{\gamma ,\chi }v,v{\mathbb{D}}_{M,x}^{\gamma ,\chi }v,v{\mathbb{D}}_{M,t}^{\gamma ,\chi }v,{\mathbb{D}}_{M,x}^{\gamma ,\chi }{\mathbb{D}}_{M,t}^{\gamma ,\chi }v,{\mathbb{D}}_{M,t}^{\gamma ,\chi }{\mathbb{D}}_{M,t}^{\gamma ,\chi }v,{\mathbb{D}}_{M,x}^{\gamma ,\chi }{\mathbb{D}}_{M,x}^{\gamma ,\chi }v,\dots )=0,$$

(3)

where $\mathcal{F}$ stands for a polynomial of v, $v=v(x,t),0<\gamma \le 1,\chi >0$. Now, the chronological approach of the suggested procedure is as follows:

Activity-1: Here, we choose the wave transformation, which is a junction between the autonomous variables and the dependent one,

$$v(x,t)=V(\psi ),\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma }),$$

(4)

where $\zeta$ stands for the parameter as well as the coefficient of space variable x and $\xi$ indicates the speed of the wave. When we implement the modification in Equation (4) in Equation (3), Equation (3) might be changed into an ordinary differential equation (ODE) as follows:

$${\mathcal{F}}_1(V,V{V}^{\prime },V{V}^{″},V^{\prime },V^{″},V^{\prime }{V}^{″},\dots )=0,$$

(5)

where ${\mathcal{F}}_1$ stands for the polynomial function of $V(\psi )$ along with its numerous ordinary derivatives.

Activity-2: We hypothesize that the following polynomial could be the solution of Equation (5):

$$V(\psi )=\sum _{m=0}^M{n}_m{({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})}^m.$$

(6)

In the above function, M is the degree of the polynomial, which must be determined by the homogeneous balance rule. Moreover, the coefficients $n_m(m=0,1,2,\dots ,M)$ can be identified with the aid of the set of algebraic equations appearing in the relevant technique. In addition, $G(\psi )$ maintains the following ODE:

$${\displaystyle \frac{d^{2}G}{d{\psi }^2}}+S{\displaystyle \frac{dG}{d\psi }}+DG+AD=0,$$

(7)

where $S,D$, and A represent the parameters. If we solve Equation (7), we will obtain the following two situations, in which $P_1$ and $P_2$ represent the constants of coefficients:

Situation 1: When $N=S^2-4D>0$,

$$({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})={\displaystyle \frac{P_1(S+\sqrt{N})+P_2(S-\sqrt{N})e^{\sqrt{N}\psi }}{P_1(S+\sqrt{N}-2)+P_2(S-\sqrt{N}-2)e^{\sqrt{N}\psi }}}.$$

Situation 2: When $N=S^2-4D<0$,

$$({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})={\displaystyle \frac{sin(\frac{\sqrt{-N}\psi }{2})(S{P}_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})(S{P}_1-P_2\sqrt{-N})}{sin(\frac{\sqrt{-N}\psi }{2})((S-2)P_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})((S-2)P_1-P_2\sqrt{-N})}}.$$

Activity-3: Combining Equation (6) with Equation (5), one can obtain the polynomial $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$. Equating the coefficients from both sides of the required polynomial, the results provide a series of algebraic equations of $A,S,D,\zeta$, and $\xi$. The probable outcomes of the NFPDE can be attained by using the value of $n_m$ and M in Equation (6).

4. Specification of the Model Equation

The M-fractional Kuralay-II equation is attributed to Professor Kuralay Yeskermessova. This equation provides a powerful framework for modeling and analyzing systems with complex, non-local, and memory-dependent dynamics across a wide range of disciplines, including viscoelastic materials, anomalous diffusion processes, electrical circuits, pharmacokinetics, economic dynamics, control systems, seismic wave propagation, and image and signal denoising. At this position, we write down the equation [41,45]:

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }v+{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }v)-uv=0,$$

(8)

$${\mathbb{D}}_{M,x}^{\gamma ,\chi }u-2\delta {\mathbb{D}}_{M,t}^{\gamma ,\chi }{\left|v\right|}^2=0,$$

(9)

where $v=v(x,t)$ stands for the complex valued function and $u=u(x,t)$ gives the real valued function. $\overline{v}$ refers to the conjugate of v, $\delta =\pm 1$, and x and t are the real variables. The suggested model has a couple of variations.

4.1. The M-Fractional Kuralay-IIA Equation

The stated model, which is integrable, has the ensuing form [41,45]:

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }v-{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }v)-uv=0.$$

(10)

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }w+{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }w)+uw=0.$$

(11)

$${\mathbb{D}}_{M,x}^{\gamma ,\chi }u+2b^2{\mathbb{D}}_{M,t}^{\gamma ,\chi }(wv)=0.$$

(12)

4.2. The M-Fractional Kuralay-IIB Equation

Another form of the Kuralay-II equation that is also integrable is as follows [41,45]:

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }v+{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }v)-uv=0.$$

(13)

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }w-{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }w)+uw=0.$$

(14)

$${\mathbb{D}}_{M,t}^{\gamma ,\chi }u-2{\mathbb{D}}_{M,x}^{\gamma ,\chi }(wv)=0.$$

(15)

5. Implementation of the Relevant Technique

In this part of this research work, we implement the selected method, which is direct as well as convenient for establishing the novel wave solutions of the KIIA equation and the KIIB equation.

5.1. The KIIA Equation

First of all, we pick $w=\delta \overline{v}$ with $b=1$ for the M-fractional Kuralay-IIA equation; then one has

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }v-{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }v)-uv=0.$$

(16)

$${\mathbb{D}}_{M,x}^{\gamma ,\chi }u-2\delta {\mathbb{D}}_{M,t}^{\gamma ,\chi }{(|v|}^2)=0.$$

(17)

Applying the wave transformation

$$v(x,t)=V(\psi )\times exp(i({\displaystyle \frac{\Gamma (1+\chi )}{\gamma }})(\kappa x^{\gamma }+\rho t^{\gamma })),u(x,t)=u(\psi ),\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma })$$

(18)

having parameters $\kappa ,\rho ,\zeta ,$ and $\xi$ to Equation (16) and Equation (17) yields two parts—one is the real part and the other is the imaginary part—as follows:

$$2\xi V^3\delta -V\zeta (\rho (\kappa -1)-c)+\xi {\zeta }^2{V}^{″}=0.$$

(19)

$$(\xi -\kappa \xi -\rho \zeta )V^{\prime }=0.$$

(20)

The speed of the wave can be calculated from Equation (20) as follows:

$$\xi ={\displaystyle \frac{\rho \zeta }{1-\kappa }}.$$

(21)

Linking Equations (19) and (21), we have

$$2\rho \delta V^3-(1-\kappa )(\rho (\kappa -1)-c)V+\rho {\zeta }^2{V}^{″}=0.$$

(22)

The homogeneous balance rule is applied between the highest-order derivative and the highest nonlinear term in Equation (22), giving $M=1$. The result of Equation (22) is thus

$$V(\psi )=n_0+n_1({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}}).$$

(23)

The constants $n_0,n_1,c$, and $\delta$ must be determined. Thus, we have

$$\begin{array}{c}\hfill n_0=n_0,n_1=-2n_0{\displaystyle \frac{(S-D-1)}{S-2D}},\delta =-{\displaystyle \frac{{(S-2D)}^2{\zeta }^2}{4n_0^2}},c={\displaystyle \frac{2\rho -4\rho \kappa +2\rho {\kappa }^2-S^2\rho {\zeta }^2+4D\rho {\zeta }^2}{2(-1+\kappa )}}.\end{array}$$

The closed-form wave solution is composed of the computed value of the constants:

Setting 1: While $N=S^2-4D>0$,

$$V(\psi )=n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{P_1(S+\sqrt{N})+P_2(S-\sqrt{N})e^{\sqrt{N}\psi }}{P_1(S+\sqrt{N}-2)+P_2(S-\sqrt{N}-2)e^{\sqrt{N}\psi }}}\right).$$

(24)

To obtain the closed-form solution of the M-fractional Kurally-IIA equation, we apply the wave transformation in Equation (18) to Equation (24):

$$\begin{array}{c}\hfill v(x,t)=\left(n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{P_1(S+\sqrt{N})+P_2(S-\sqrt{N})e^{\sqrt{N}\psi }}{P_1(S+\sqrt{N}-2)+P_2(S-\sqrt{N}-2)e^{\sqrt{N}\psi }}}\right)\right)\\ \hfill \times exp(i({\displaystyle \frac{\Gamma (1+\chi )}{\gamma }})(\kappa x^{\gamma }+\rho t^{\gamma })),\end{array}$$

(25)

where $\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma })$.

Setting 2: While $N=S^2-4D<0$,

$$V(\psi )=n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{sin(\frac{\sqrt{-N}\psi }{2})(S{P}_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})(S{P}_1-P_2\sqrt{-N})}{sin(\frac{\sqrt{-N}\psi }{2})((S-2)P_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})((S-2)P_1-P_2\sqrt{-N})}}\right).$$

(26)

The closed-form solution of the M-fractional Kurally-IIA equation can be evaluated with the aid of the wave transformation in Equation (18) by use of Equation (26):

$$\begin{array}{c}\hfill v(x,t)=\left(n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{sin(\frac{\sqrt{-N}\psi }{2})(S{P}_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})(S{P}_1-P_2\sqrt{-N})}{sin(\frac{\sqrt{-N}\psi }{2})((S-2)P_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})((S-2)P_1-P_2\sqrt{-N})}}\right)\right)\\ \hfill exp(i({\displaystyle \frac{\Gamma (1+\chi )}{\gamma }})(\kappa x^{\gamma }+\rho t^{\gamma })),\end{array}$$

(27)

where $\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma })$.

5.2. The KIIB Equation

In the same way, if we choose $w=\delta \overline{v}$ for the M-fractional Kuralay-IIB equation, then we have

$$i{\mathbb{D}}_{M,t}^{\gamma ,\chi }v+{\mathbb{D}}_{M,x}^{\gamma ,\chi }({\mathbb{D}}_{M,t}^{\gamma ,\chi }v)-uv=0.$$

(28)

$${\mathbb{D}}_{M,t}^{\gamma ,\chi }u+2\delta {\mathbb{D}}_{M,x}^{\gamma ,\chi }{(|v|}^2)=0.$$

(29)

If we implement the modification given in Equation (18) in Equation (28) and Equation (29), then the parts, including the real part and imaginary part, are given as follows:

$$2V^3\zeta \delta -\xi V(\rho (\kappa +1)-c)+{\xi }^2\zeta V^{″}=0.$$

(30)

$$(\xi +\kappa \xi +\rho \zeta )V^{\prime }=0.$$

(31)

The speed of the wave can be evaluated using Equation (31) as follows:

$$\xi =-{\displaystyle \frac{\rho \zeta }{1+\kappa }}.$$

(32)

Combining Equation (30) together with Equation (32), one can write the following:

$$2{(1+\kappa )}^2\delta V^3+(1+\kappa )({\rho }^2(\kappa +1)-\rho c)V+{\rho }^2{\zeta }^2{V}^{″}=0.$$

(33)

Utilizing the homogeneous balance rule in Equation (33), we can set $M=1$. The solution to Equation (33) is as follows:

$$V(\psi )=n_0+n_1({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}}).$$

(34)

We have to compute the constants $n_0,n_1,c$, and $\delta$. Subsequently, a productive execution provides us with

$$n_0=n_0,n_1=-2n_0{\displaystyle \frac{(S-D-1)}{S-2D}},\delta =-{\displaystyle \frac{{(S-2D)}^2{\rho }^2{\zeta }^2}{4{(1+\kappa )}^2{n}_0^2}},c={\displaystyle \frac{2\rho +4\rho \kappa +2\rho {\kappa }^2-S^2\rho {\zeta }^2+4D\rho {\zeta }^2}{2(1+\kappa )}}.$$

Combining the closed-form wave solution with the value of the constants, we have the following.

Setting 1: While $N=S^2-4D>0$,

$$V(\psi )=n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{P_1(S+\sqrt{N})+P_2(S-\sqrt{N})e^{\sqrt{N}\psi }}{P_1(S+\sqrt{N}-2)+P_2(S-\sqrt{N}-2)e^{\sqrt{N}\psi }}}\right).$$

(35)

To generate the closed-form solution of the M-fractional Kurally-IIB equation, we utilize the wave transformation in Equation (18) within Equation (35):

$$\begin{array}{c}\hfill v(x,t)=\left(n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{P_1(S+\sqrt{N})+P_2(S-\sqrt{N})e^{\sqrt{N}\psi }}{P_1(S+\sqrt{N}-2)+P_2(S-\sqrt{N}-2)e^{\sqrt{N}\psi }}}\right)\right)\\ \hfill \times exp(i({\displaystyle \frac{\Gamma (1+\chi )}{\gamma }})(\kappa x^{\gamma }+\rho t^{\gamma })),\end{array}$$

(36)

where $\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma })$.

Setting 2: While $N=S^2-4D<0$,

$$V(\psi )=n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{sin(\frac{\sqrt{-N}\psi }{2})(S{P}_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})(S{P}_1-P_2\sqrt{-N})}{sin(\frac{\sqrt{-N}\psi }{2})((S-2)P_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})((S-2)P_1-P_2\sqrt{-N})}}\right).$$

(37)

Performing the wave transformation given in Equation (18) in Equation (37), one may compose the closed-form solution of the M-fractional Kurally-IIB equation:

$$\begin{array}{c}\hfill v(x,t)=\left(n_0-2n_0\left({\displaystyle \frac{S-D-1}{S-2D}}\right)\left({\displaystyle \frac{sin(\frac{\sqrt{-N}\psi }{2})(S{P}_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})(S{P}_1-P_2\sqrt{-N})}{sin(\frac{\sqrt{-N}\psi }{2})((S-2)P_2+P_1\sqrt{-N})+cos(\frac{\sqrt{-N}\psi }{2})((S-2)P_1-P_2\sqrt{-N})}}\right)\right)\\ \hfill \times exp(i({\displaystyle \frac{\Gamma (1+\chi )}{\gamma }})(\kappa x^{\gamma }+\rho t^{\gamma }))\end{array}$$

(38)

where $\psi ={\displaystyle \frac{\Gamma (1+\chi )}{\gamma }}(\zeta x^{\gamma }+\xi t^{\gamma })$.

6. Some Analyses of the Model Equations

The chaotic character as well as sensitivity of the KIIA and KIIB equations are two aspects that we are interested in investigating in the next section.

6.1. Chaotic Nature of KIIA Equation

A dynamical system is attained by applying the Galilean transformation given in Equation (22) as follows:

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV}{d\psi }}\end{array}& =M\hfill \\ \hfill {\displaystyle \frac{dM}{d\psi }}& =W_1{V}^3+W_{2}V\hfill \end{array},$$

(39)

wherein $W_1={\displaystyle \frac{-2\delta }{{\zeta }^2}}$ and $W_2={\displaystyle \frac{(1-\kappa )(\rho (\kappa -1)-c)}{\rho {\zeta }^2}}$. Now, we introduce a perturbation term ${\beta }_{1}cos({\beta }_{2}t)$, where ${\beta }_1$ means amplitude and ${\beta }_2$ means frequency, in Equation (39) as follows:

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV(t)}{dt}}\end{array}& =M\hfill \\ \hfill {\displaystyle \frac{dM(t)}{dt}}& =W_{1}V{(t)}^3+W_{2}V(t)+{\beta }_{1}cos({\beta }_{2}t)\hfill \end{array}.$$

(40)

In this section, we examine the influence of the enhanced term in the system given in Equation (43), as provided in Figure 1, Figure 2 and Figure 3. Then, we set the specific values for each parameter in Figure 1a as follows: $\delta =1,\zeta =\sqrt{{\displaystyle \frac{2}{3}}},\rho =3,\kappa =6,c=13,{\beta }_1=0.89,{\beta }_2={\displaystyle \frac{\pi }{2}}$. In Figure 1b, the values are $\delta =1,\zeta =\sqrt{{\displaystyle \frac{2}{3}}},\rho =3,\kappa =6,c=13,{\beta }_1=2.89,{\beta }_2={\displaystyle \frac{\pi }{2}}$. In Figure 1c, the values are $\delta =1,\zeta =\sqrt{{\displaystyle \frac{2}{3}}},\rho =3,\kappa =6,c=13,{\beta }_1=0.1,{\beta }_2={\displaystyle \frac{\pi }{2}}$. Butterfly-shape dynamics, ringlet dynamics, complex dynamics, periodic dynamics, and scroll dynamics are established in this investigation, all of which have multiple uses in nonlinear studies. In the study of nonlinear dynamics, these play significant roles across many scientific and engineering fields, including weather modeling, secure communication systems, prediction and analysis in climate systems, robotics motion planning with periodic behavior, study of population dynamics in ecology, nonlinear optics and laser systems, financial market modeling, heartbeat and respiratory cycle modeling, and simulation of fluid turbulence and plasma systems.

6.2. Sensitivity Analysis of KIIA Equation

This part will focus on the sensitivity analysis of the KIIA equation, which will be discussed later. We apply the well-known and efficient Runge–Kutta process to investigate the sensitivity of the dynamical system given by Equation (39). To test the system’s reliability, we solve the following dynamical system using the Runge–Kutta scheme:

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV(t)}{dt}}\end{array}& =M\hfill \\ \hfill {\displaystyle \frac{dM(t)}{dt}}& =W_{1}V{(t)}^3+W_{2}V(t)\hfill \end{array}.$$

(41)

At this point, we choose the values of the parameters as $\delta =1,\zeta =\sqrt{2},\kappa =2,$$\rho =1,c=4$, together with the following initial conditions:

• $V(0)=0.1$ and $M=0$
• $V(0)=0$ and $M=0.1$
• $V(0)=0.2$ and $M=0$
• $V(0)=0$ and $M=0.2$.

In Figure 4, orange curves illustrate the dynamics of class M, while the blue curves show the dynamics of class V. A detailed analysis of the graphs indicates that even slight alterations in the initial parameters lead to significant changes in the behavior of the system. This aids decision-making in engineering, physics, and biology by understanding potential deviation ranges in real-world applications.

6.3. Chaotic Nature of KIIB Equation

If we apply the Galilean transformation in Equation (33), we have a dynamical system as follows:

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV}{d\psi }}\end{array}& =N\hfill \\ \hfill {\displaystyle \frac{dN}{d\psi }}& =W_3{V}^3+W_{4}V\hfill \end{array},$$

(42)

where $W_3={\displaystyle \frac{-2{(1+\kappa )}^2\delta }{{(\rho \zeta )}^2}}$ and $W_4={\displaystyle \frac{-(1+\kappa )({\rho }^2(\kappa +1)-\rho c)}{{(\rho \zeta )}^2}}$. To accomplish this analysis, we add a perturbation term ${\beta }_{3}cos({\beta }_{4}t)$, with ${\beta }_3$ and ${\beta }_4$ as the amplitude and frequency, respectively, as given in Equation (42) as

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV(t)}{dt}}\end{array}& =N\hfill \\ \hfill {\displaystyle \frac{dN(t)}{dt}}& =W_{3}V{(t)}^3+W_{4}V(t)+{\beta }_{3}cos({\beta }_{4}t)\hfill \end{array}.$$

(43)

Now, we consider the values of the parameters in Figure 5a ($\delta =1,\kappa =-3,\rho =1,\zeta =2,c=4,{\beta }_3=3.5,{\beta }_4={\displaystyle \frac{\pi }{2}}$), in Figure 5b ($\delta =1,\kappa =-3,\rho =1,\zeta =2,c=4,{\beta }_3=0.24,{\beta }_4={\displaystyle \frac{\pi }{2}}$), in Figure 5c ($\delta =1,\kappa =-3,\rho =\sqrt{2},\zeta =2,c=8\sqrt{2},{\beta }_3=0.1,{\beta }_4={\displaystyle \frac{\pi }{2}}$), and the others. Upon close inspection, we detect the complex dynamics, ringlet dynamics, periodic dynamics, strange dynamics, and scroll dynamics of the selected dynamical system, which are illustrated in Figure 5, Figure 6 and Figure 7. It is possible that mathematical physics, along with contemporary engineering, may benefit from expected, specified, powerful, and readily apparent chaotic instances. For more clarifications of chaos, some quantitative indicators of chaos, such as Lyapunov exponents, bifurcation diagrams, etc., are effectively discussed in [54,55,56]. In both systems of Equations (40) and (41), we have seen that they seem to have a chaotic behavior, which must be proved in future work.

6.4. Sensitivity Analysis of KIIB Equation

Here, we are going to take a look at the KIIB equation’s sensitivity assessment. Using the renowned and powerful Runge–Kutta procedure, we analyze the sensitivity of the dynamical system defined by Equation (42). The following dynamical system has been resolved using the Runge–Kutta technique in order to monitor how sensitive it is:

$$\{\begin{array}{cc}\hfill \begin{array}{c}\hfill {\displaystyle \frac{dV(t)}{dt}}\end{array}& =N\hfill \\ \hfill {\displaystyle \frac{dN(t)}{dt}}& =W_{3}V{(t)}^3+W_{4}V(t)\hfill \end{array}.$$

(44)

Here, we consider the parameters a$\delta =1,\zeta =2,\kappa =1,\rho =\sqrt{2},c=-16\sqrt{2}$, along with the initial conditions

• $V(0)=0.1$ and $N=0$
• $V(0)=0$ and $N=0.1$
• $V(0)=0.2$ and $N=0$
• $V(0)=0$ and $N=0.2$.

From Figure 8 we see that the dynamics of class V are represented by blue curves, while the dynamics of class N are shown by orange curves. When the figures are closely examined, it becomes clear that even small changes in the initial configuration have a significant impact on how the system behaves. Understanding possible deviation ranges in practical applications helps engineers, physicists, and biologists make better decisions.

7. Graphs of Soliton Solutions with Their Physical Interpretation

Traveling wave solutions provide indispensable insights into the activities of waves in miscellaneous contexts, aiding in the development of technologies and systems that harness wave propagation for practical applications. We have applied the suggested method to produce wave solutions associated with the considered NFPDE, and in the next paragraph, we visually analyze their intricate nature. Here, we notice the kink-shape, singular periodic-shape, anti-kink-shape, and singular kink-shape solitons of the established closed-form solutions. This section will provide insight into the unique characteristics and purposes of these closed-form solutions. Our obtained findings may provide insight into the stated model and will help further develop the explored model. We depict the $3D$, contour, and $2D$ formats of the identical equation, which are placed in sub-figures (a), (b), and (c) of Figure 9, Figure 10, Figure 11 and Figure 12, respectively. First, we display the closed-form solution in Equation (25) in three ways—$3D$ and contour shape with $x,t\in [0,10]$, and $2D$ with $x\in [0,10]$, together with the parameters $n_0=1,\gamma =0.95,\chi =1,\zeta =-1,\xi =1,S=2,D=0.75,P_1=-1,P_2=2$, and $t=2$, which show the kink shape soliton in Figure 9. And in Figure 10, we provide the closed-form solution in Equation (27) in three formats—$3D$ and contour shape with $x,t\in [0,30]$, and $2D$ with $x\in [0,50]$, with the parameters $n_0=1,\gamma =0.95,\chi =1,\zeta =0.5,\xi =0.5,S=0.001,D=0.3,P_1=0.001,P_2=0.001$, and $t=1$, which are associated with the singular periodic shape.

Now, the anti-kink shape is depicted for the solution in Equation (36) in three forms—$3D$ and contour shape with $x,t\in [0,10]$, and $2D$ with $x\in [0,10]$, together with the parameters $n_0=-1,\gamma =0.95,\chi =1,\zeta =-1,\xi =1,S=7,D=0.02,P_1=1,P_2=-2$, and $t=2$ (Figure 11). Before we finish this section, we display the findings of Equation (38) in three formats—$3D$ shape with $x,t\in [0,15]$, contour shape with $x,t\in [0,20]$, and $2D$ with $x\in [0,20]$, with the parameters $n_0=-1,\gamma =0.95,\chi =1,\zeta =0.2,\xi =0.2,S=0.001,$$D=0.3,P_1=5,P_2=3$, and $t=2$, representing the singular kink-shape soliton in Figure 12.

Sagidullayeva et al. [40] found the one-soliton solutions of the relevant model. Zafar et al. [41] determined the dark, bright, dark-bright, and singular soliton solutions of the KIIE. In [45], periodic, solitary, and trigonometric solutions of the stated model are discovered. Faridi et al. [42] explored the rational, mixed periodic, mixed trigonometric, and mixed hyperbolic solutions of the KIIE. But it is evident from all the figures that the discoveries of the aforementioned model remain remarkably novel and have not been extensively examined in prior studies. We obtained two new solutions for KIIA and two new ones for KIIB and we can also derive more particular solutions using special values for the parameters. These findings will help scientists gain a better grasp of the most intriguing properties of NFPDEs, which lie behind the various marvels of nonlinear science and technology.

An examination of the differences and similarities between the outcomes achieved for the M-fractional Kuralay-IIA equation and the M-fractional Kuralay-IIB equation is provided in the following

Table 1. Results comparison of the M-fractional Kuralay-IIA and Kuralay-IIB equations.

Equation	Outcome(s)
The M-fractional Kuralay-IIA equation	Kink-shape, and singular priodic-shape solitons
The M-fractional Kuralay-IIB equation	Anti-kink-shape and singular kink-shape solitons

.

In the next section, we compare our findings with existing findings. Based on the comparison table with other procedures given in

Table 2. Comparison of our results with existing results.

Author(s)	Method(s)	Outcomes	Analyses
Arafat et al. [43]	The extended hyperbolic function method and the improved F-expansion method	Periodic soliton, kink soliton, bell soliton, and dark soliton	Bifurcation analysis
Rashedi et al. [52]	The Riccati–Bernoulli sub-ODE method combined with the Bäcklund transformation	Periodic perturbation, persistent periodicity steady oscillations, and distant periodical peaks	None
Zafar et al. [41]	The $ex{p}_a$ function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes	Flat kink soliton and singular kink soliton	None
Our research work	The $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach	Anti-kink, singular periodic, kink, and singular kink-shape solitons	Chaotic analysis and sensitivity analysis

, we note the originality of our findings.

8. Conclusions

This study effectively provided some noteworthy analyses, a chaotic analysis, and a sensitivity assessment of the M-fractional Kuralay-II equation (KIIA and KIIB equation). Butterfly-shape dynamics, ringlet dynamics, complex dynamics, periodic dynamics, strange dynamics, scroll dynamics, and other practically meaningful features were all explored in the chaotic analysis. We performed a sensitivity analysis of the models and noticed that the results of these models are extremely responsive to their initial state. Closed-form wave solutions are substantially relevant in acoustics, microwave engineering, optical fibers, electromagnetics, coastal engineering, modern physics, electrical engineering, quantum mechanics, medical sciences, meteorology, and signal processing because they delineate a wide range of phenomena, from sound waves and light waves to water waves and even seismic waves. Our investigation using the $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach enabled us to achieve new closed-form wave solutions for the selected models that have not been examined in the literature before. We investigated the chaotic behaviours and conducted a sensitivity analysis of these two models, which are given in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. In addition, the new solutions that were found belong to the rational, exponential, and trigonometric function forms, which are sophisticated and completely distinct from the solutions in the existing literature. The observed results are displayed in Figure 9, Figure 10, Figure 11 and Figure 12 in $3D$, contour, and $2D$ formats, which have singular periodic, anti-kink, kink, and singular kink-shape solitons. Since they offer a deeper understanding of this model, these closed-form wave solutions seem extremely appealing. The outcomes reported in this research could help further improve our understanding of the aforementioned model applied to anomalous diffusion processes, electrical circuits, complex biological systems, economic dynamics, control systems, seismic wave propagation, nonlinear optics, groundwater flow, image and signal denoising, population dynamics, and ferromagnetic materials. We are optimistic that our strategy can be utilized to discover novel and intriguing closed-form wave solutions for a wide variety of NFPDEs, including the sine-Gordon equation, complex fluid flow equations, the Benjamin–Bona–Mahony equation, the Boussinesq equation, nonlinear Schrödinger-type equations, the modified Korteweg–de Vries equation, and numerous systems in Bose–Einstein condensates that occur throughout mathematical physics and contemporary engineering sciences.