Analytical Study of Soliton Solutions and Modulation Instability Analysis in the M-Truncated Fractional Coupled Ivancevic Option-Pricing Model

Abstract

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management.

1. Introduction

In the current era, the economy and finance have arisen as some of the most studied fields worldwide. With the development of advanced scientific and technical instruments, scholars and scientists can produce increasingly sophisticated models and products [1]. These tools are essential for determining optimal conditions in various aspects of daily life, particularly in the context of shipping and logistics. The initial step involves constructing mathematical models, whether they are complex-valued or real-valued, using wave functions to understand these systems. Many such models have been developed to analyze current and future trends in economic and financial markets [2]. Soliton theory is widely applied in these models as it provides exact solutions such as periodic, singular, dark, bright and traveling wave solutions. As a result, a number of mathematical models have been created to investigate and forecast wave dynamics in both present and future economic situations. A popular method in this regard is soliton theory, which offers important insights into a variety of wave solutions, including periodic, singular, dark, bright and moving waves. For comprehending, forecasting and managing intricate systems in production and other economic activities, these dynamic solutions provide vital information. In their study of backward stochastic differential equations (SDEs) in finance, Sharp et al. investigated the stability of SDEs under small perturbations of coefficients and boundary conditions [3,4]. Their study proved that these equations existed and that they had special properties. Similarly, Irving and his colleagues investigated mixed linear-nonlinear coupled differential equations utilizing multivariate discrete time series sequences [5]. Jin et al. demonstrated optimal consumption and portfolio rules in a continuous-time finance model [6]. More recently, Ganesh et al. investigated fractional-order impulsive stochastic differential equations in controllability, demonstrating the theory in numerical integration using the Haar wavelet approximation approach [7]. Furthermore, Decardi-Nelson and his group produced a strong economic model that introduced the idea of risk variables in controller design and Samarskii suggested economic splitting techniques [8,9]. They offered an algorithm for effectively monitoring economic zones. To solve complicated dynamical systems, Adomian used the decomposition approach to model and analyze a national economy utilizing coupled nonlinear stochastic multidimensional operator equations (both discrete and continuous) [10]. With the growth of computational power, the modeling of large-scale systems such as power plants has also been extensively studied [11].

Due to the inherent nonlinearity of physical phenomena, mathematical modeling is generally the most effective representation method [12,13,14]. It may now employ partial differential equations to thoroughly comprehend and delineate the attributes of physical challenges [15,16,17]. The primary physical challenge for these models is obtaining mathematical explanations for propagating waves. Various disciplines, such as physics, geophysics, climatic dynamics, economics, biology and others, now extensively utilize randomness [18,19,20,21]. Solutions to a variety of nonlinear evolution equations (NLEEs) have been developed in nonlinear applied science domains, including geology, solid-state physics, chemical physics and optical fiber. The solution of NLEEs has been investigated in a variety of scientific and engineering disciplines. There are a multitude of technical and mathematical subjects that are addressed, such as ocean propagation, hydrodynamics, thermal capacity, magnetism, quantum dynamics and seismic waves [22,23]. It is essential to establish analytical approaches for these NLEEs and demonstrate a comprehensive understanding of the qualitative characteristics of these scenarios. Various scholars have exhibited a keen interest in identifying an exact solution to these NLEEs. It has been demonstrated that employing symbolic tools yields numerous suitable and effective methods for identifying optimal solutions to various NLEEs. The unified Riccati equation-expansion scheme [24], the modified Fan-sub equation technique [25], the hyperbolic function method [26], the Hirota method [27], the Gordon expansion scheme [28] and the Kudryashov method [29] represent merely a subset of the several practical methodologies accessible.

The Ivancevic option-pricing model (IOPM) [30] is a widely researched nonlinear partial differential equation

$$\begin{array}{c}\hfill i{D}_{M,t}^{\alpha ,\beta }\phi +{\displaystyle \frac{1}{2}}{D}_{M,s}^{2\alpha ,\beta }\phi +{\mho \left(\right|\phi |}^2+{\left|\gamma \right|}^2)\phi =0,\\ \hfill i{D}_{M,t}^{\alpha ,\beta }\gamma +{\displaystyle \frac{1}{2}}{D}_{M,s}^{2\alpha ,\beta }\gamma +{\mho \left(\right|\phi |}^2+{\left|\gamma \right|}^2)\gamma =0,\end{array}$$

(1)

where $D_M^{\alpha ,\beta }$ denotes the M-truncated derivative with $\alpha \in (0,1]$ and $\beta >0$, while $\phi =\phi (s,t)$ is the volatility, $\gamma =\gamma (s,t)$ is the option price and $\mho (r,\sigma )=r{\sum }_{h=1}^n{\sigma }_h{g}_h,$ for ${\sigma }_h=-{\sigma }_h+c\left|\phi \right|g_h\left|\gamma \right|$ [31], is the adaptive market potential. The adaptive marketheat-potential is proportional to the dot product of the Gaussian kernel vector $g_h$ and the synaptic weight vector ${\sigma }_h$. As the Gaussian kernel vector cannot be either zero or negative, the product may be zero, positive, or negative based on the synaptic weight vector’s value. It is possible to dynamically change the weight. on a number of variables that impact the general behavior and dynamics of the market, such as past data, current market conditions and outside events. The model’s adaptive market potential incorporates Gaussian kernels to capture smoothed, non-local price influences, while the synaptic weights explicitly encode cross-asset correlations. These components are calibrated to historical data, with dimensionless units ensuring scalability. To adhere to financial principles, domain transforms are applied to enforce no-arbitrage and positivity constraints. The bidirectional associative memory model Equation (1) utilizes quantum neural computation, providing a spatial and temporal extension to Kosko’s BAM neural network family [32]. Additionally, phenomena like the propagation of volatility/prices and the interplay of shock and solitary waves may be captured by the properties of shock waves and solitary waves in the coupled nonlinear Schrodinger (NLS) equations. When $\sigma$−learning is not present (that is, for a fixed $\mho =r$, which stands for the interest rate), the coupled NLS equations basically characterize the well-known Manakov system, which was created by S. Manakov in 1973 [33]. It has been demonstrated that this system is fully integrable since there are an infinite number of involutive integrals of motion. The “bright” and “dark” solutions are among its features.

Fractional calculus, initially developed for expressing the concepts of non-integer variations and integrals in particular, offers an extensive mathematics framework for elucidating different phenomena across diverse scientific fields [34,35]. This growing significance is driven by an increasing need for accurate simulations of both historic and contemporary physical processes [36,37]. Research has shown the efficacy of fractional operators in representing phenomena from nature, indicating that fractional-order models outperform traditional non-integer systems in both efficiency and effectiveness [38,39].

Nevertheless, a comprehensive review of the literature has revealed that the Kumar–Malik technique [40], the modified Sardar sub-equation approach [41] and the modified Arnous approach [42] have not been used to solve the governing Equation (1). This paper’s contribution is highly relevant in a number of ways. First of all, it expands the analytical toolbox for resolving fractional NLEEs, which are not limited to financial modeling but are utilized in many scientific domains. Second, by focusing on option pricing, a vital aspect of financial markets, the study addresses a pertinent problem that real-world traders and investors face. Third, the incorporation of M-truncated fractional derivatives enhances the model’s intricacy and improves its representation of option price dynamics and volatility.

The following sections are included in this article: Section 2 consists of the truncated M-fractional derivative. The extraction of solutions in Section 3 and Section 4 investigate the modulation instability analysis. Discussion is presented in Section 5, while the concluding remarks are given in Section 6.

2. The Truncated $\mathit{M}$-Fractional Derivative

Recent research has demonstrated that, especially in nonlinear systems, the truncated M-fractional derivative offers a more accurate depiction of the behavior of solitary waves than other fractional derivatives [43]. This phenomenon exhibits a wide range of practical applications across various scientific disciplines, including engineering, signal processing and electromagnetics. The truncated M-fractional derivative is a significant parameter as it accurately characterizes complex systems exhibiting memory effects, long-range and nonlocal interactions and nonlinear dynamics [44].

Definition 1. 

Let $k:[0,\infty )\to \mathbb{R}$, then the new truncated M-fractional derivative of k of order β is discussed [45] as:

$${\mathcal{D}}_M^{\beta ,\zeta }\left\{\left(k\right)\left(t\right)\right\}=\underset{\beta \to 0}{lim}{\displaystyle \frac{k\left(t{\mathbb{E}}_{\zeta }\left(\beta t^{1-\beta }\right)\right)-k\left(t\right)}{\beta }},\forall t>0,0<\beta <1,\zeta >0,$$

(2)

where ${\mathbb{E}}_{\zeta }(.)$ is a truncated Mittag-Leffler function of one parameter [45].

Theorem 1. 

Let $0<\beta \le 1$, $\zeta >0$, $p,s\in \mathbb{R}$ and $f,k$ β-differentiable at a point $t>0$ [46]. Then:

• ${\mathcal{D}}_M^{\beta ,\zeta }\left\{(pf+sk)\left(t\right)\right\}=p{\mathcal{D}}_M^{\beta ,\zeta }\left\{f\left(t\right)\right\}+s{\mathcal{D}}_M^{\beta ,\zeta }\left\{k\left(t\right)\right\}$.
• ${\mathcal{D}}_M^{\beta ,\zeta }\left\{(f.k)\left(t\right)\right\}=f\left(t\right){\mathcal{D}}_M^{\beta ,\zeta }\left\{k\left(t\right)\right\}+k\left(t\right){\mathcal{D}}_M^{\beta ,\zeta }\left\{f\left(t\right)\right\}$.
• ${\mathcal{D}}_M^{\beta ,\zeta }\left\{{\displaystyle \frac{f}{k}}\left(t\right)\right\}={\displaystyle \frac{k\left(t\right){\mathcal{D}}_M^{\beta ,\zeta }\left\{f\left(t\right)\right\}-f\left(t\right){\mathcal{D}}_M^{\beta ,\zeta }\left\{k\left(t\right)\right\}}{{\left[k\left(t\right)\right]}^2}}$.
• ${\mathcal{D}}_M^{\beta ,\zeta }\left\{f\left(t\right)\right\}=0$, where $f\left(t\right)=c$ is a constant.
• If $f\left(t\right)$ is differentiable, then ${\mathcal{D}}_M^{\beta ,\zeta }\left\{f\left(t\right)\right\}={\displaystyle \frac{t^{1-\beta }}{\Gamma (\zeta +1)}}{\displaystyle \frac{df\left(t\right)}{dt}}.$

3. Extraction of the Solutions

Assume the following traveling wave transformation as: $\phi (s,t)=\mathsf{\Phi }(\xi )e^{i\chi }\mathrm{and}\gamma (s,t)=\mathsf{\Psi }(\xi )e^{i\chi },\mathrm{where}\xi ={\displaystyle \frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }+\tau t^{\alpha }\right)}{\alpha }}$, $\chi ={\displaystyle \frac{\Gamma (\beta +1)\left(\kappa s^{\alpha }+\omega t^{\alpha }\right)}{\alpha }}$ and $0<\alpha \le 1$. Here, $\omega$ and $\tau$ represent temporal and $\zeta$ and $\kappa$ are parameters derived from the asset price of the commodity. In system (1), we apply the transformation to obtain

Real part

$$\begin{array}{c}\hfill {\zeta }^2{\mathsf{\Phi }}^{″}(\xi )+2\mho \left({\Omega }^2+1\right){\mathsf{\Phi }}^3(\xi )-\left({\kappa }^2+2\omega \right)\mathsf{\Phi }(\xi )=0.\end{array}$$

(3)

When we take $\mathsf{\Psi }(\xi )=\Omega \mathsf{\Phi }(\xi ).$

Imaginary part

We obtain

$$\tau =-\kappa \zeta .$$

(4)

Afterward, applying the balance technique in the system (3), offers, $n=1$.

3.1. Solutions via Newly Created Kumar-Malik Method

The proposed solution takes the following form:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\xi )=A_0+\sum _{h=1}^n{A}_h{\left(Q(\xi )\right)}^h.\end{array}$$

(5)

Moreover, $Q(\xi )$ fulfils the following first order ODE:

$$\begin{array}{c}\hfill [{\left(Q^{\prime }(\xi )\right]}^2=[{\alpha }_1{Q}^4(\xi )+{\alpha }_2{Q}^3(\xi )+{\alpha }_3{Q}^2(\xi )+{\alpha }_{4}Q(\xi )+{\alpha }_5],\end{array}$$

(6)

where ${\alpha }_i(i=1,\cdots ,5)$ are constants. Moreover, the exact solutions of Equation (6) can be observed by scrutinizing four distinct solution families. For a comprehensive discussion, refer to [40]. For $n=1$, the Equation (5) simplifies to:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\xi )=A_0+A_{1}Q(\xi ).\end{array}$$

(7)

Substituting Equation (7) and its derivatives (derived from Equation (6)) into Equation (3), we gather coefficients of like powers of $Q(\xi )$. Setting these coefficients to zero and solving the resulting system using representative computation (via Mathematica) yields the subsequent solution sets.

Remark 1. 

For brevity, we introduce the following: ${\mathsf{\Sigma }}_1=4{\alpha }_1{\alpha }_3-{\alpha }_2^2,{\mathsf{\Sigma }}_2=16{\alpha }_1{\alpha }_3-5{\alpha }_2^2,{\mathsf{\Sigma }}_3=8{\alpha }_1{\alpha }_3-3{\alpha }_2^2$ for next four cases.

Case-1: For ${\alpha }_4={\displaystyle \frac{\left(4{\alpha }_1{\alpha }_3-{\alpha }_2^2\right){\alpha }_2}{8{\alpha }_1^2}},{\alpha }_5=0.$ Solving the algebraic system yields

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_0={\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},A_1={\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},\omega ={\displaystyle \frac{1}{16}}\left(\left(8{\alpha }_3-{\displaystyle \frac{3{\alpha }_2^2}{{\alpha }_1}}\right){\zeta }^2-8{\kappa }^2\right).\hfill \end{array}\right.\end{array}$$

• The Jacobi elliptic solutions
  Subcase 1.1. If ${\alpha }_1<0$ and ${\mathsf{\Sigma }}_1>0$

  $$\begin{array}{c}\hfill {\phi }_1(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{cn}\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_1}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\alpha {\alpha }_1}},{\displaystyle \frac{{\alpha }_2}{2\sqrt{{\mathsf{\Sigma }}_1}}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (8)

  $$\begin{array}{c}\hfill {\phi }_2(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{dn}\left({\displaystyle \frac{{\alpha }_2\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha \sqrt{-{\alpha }_1}}},{\displaystyle \frac{2\sqrt{{\mathsf{\Sigma }}_1}}{{\alpha }_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (9)
  Subcase 1.2. If ${\alpha }_1<0,{\mathsf{\Sigma }}_1<0\mathrm{and}{\mathsf{\Sigma }}_2<0$

  $$\begin{array}{c}\hfill {\phi }_3(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{cn}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_1}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\alpha {\alpha }_1}},{\displaystyle \frac{\sqrt{{\mathsf{\Sigma }}_1{\mathsf{\Sigma }}_2}}{2{\mathsf{\Sigma }}_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (10)

  $$\begin{array}{c}\hfill {\phi }_4(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{dn}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_2}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}},{\displaystyle \frac{2\sqrt{{\mathsf{\Sigma }}_1{\mathsf{\Sigma }}_2}}{{\mathsf{\Sigma }}_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (11)
  Subcase 1.3. If ${\alpha }_1<0,{\mathsf{\Sigma }}_1>0\mathrm{and}{\mathsf{\Sigma }}_2<0$

  $$\begin{array}{c}\hfill {\phi }_5(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{nc}\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_1}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\alpha {\alpha }_1}},{\displaystyle \frac{{\alpha }_2}{2\sqrt{{\mathsf{\Sigma }}_1}}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (12)

  $$\begin{array}{c}\hfill {\phi }_6(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{nd}\left({\displaystyle \frac{{\alpha }_2\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha \sqrt{-{\alpha }_1}}},{\displaystyle \frac{2\sqrt{{\mathsf{\Sigma }}_1}}{{\alpha }_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (13)
  Subcase 1.4. If ${\alpha }_1{\mathsf{\Sigma }}_1>0\mathrm{and}{\mathsf{\Sigma }}_1{\mathsf{\Sigma }}_2>0$

  $$\begin{array}{c}\hfill {\phi }_7(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{nc}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_1}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\alpha {\alpha }_1}},{\displaystyle \frac{\sqrt{{\mathsf{\Sigma }}_1{\mathsf{\Sigma }}_2}}{2{\mathsf{\Sigma }}_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (14)

  $$\begin{array}{c}\hfill {\phi }_8(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{nd}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_2}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}},{\displaystyle \frac{2\sqrt{{\mathsf{\Sigma }}_1{\mathsf{\Sigma }}_2}}{{\mathsf{\Sigma }}_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (15)
  Subcase 1.5. If ${\alpha }_1>0\mathrm{and}{\mathsf{\Sigma }}_2<0$

  $$\begin{array}{c}\hfill {\phi }_9(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{ns}\left({\displaystyle \frac{{\alpha }_2\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha \sqrt{{\alpha }_1}}},{\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{{\alpha }_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (16)

  $$\begin{array}{c}\hfill {\phi }_{10}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{ns}\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_2}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}},{\displaystyle \frac{{\alpha }_2}{\sqrt{-{\mathsf{\Sigma }}_2}}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (17)

  $$\begin{array}{c}\hfill {\phi }_{11}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{4{\alpha }_1}}\left(\mathrm{sn}\left({\displaystyle \frac{{\alpha }_2\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha \sqrt{{\alpha }_1}}},{\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_2}}{{\alpha }_2}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (18)

  $$\begin{array}{c}\hfill {\phi }_{12}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\left(\mathrm{sn}\left({\displaystyle \frac{\sqrt{-{\mathsf{\alpha }}_1{\mathsf{\Sigma }}_2}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}},{\displaystyle \frac{{\alpha }_2}{\sqrt{-{\mathsf{\Sigma }}_2}}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (19)
  Case-2: For ${\alpha }_4={\displaystyle \frac{{\alpha }_2{\mathsf{\Sigma }}_1}{8{\alpha }_1^2}},{\alpha }_5={\displaystyle \frac{{\alpha }_2^2{\mathsf{\Sigma }}_2}{256{\alpha }_1^3}}.$ Solving the algebraic system yields

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_0={\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},A_1={\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},\omega ={\displaystyle \frac{1}{16}}\left(\left(8{\alpha }_3-{\displaystyle \frac{3{\alpha }_2^2}{{\alpha }_1}}\right){\zeta }^2-8{\kappa }^2\right).\hfill \end{array}\right.\end{array}$$
• The explicit dark, singular and periodic wave solutions
  Subcase 2.1. If ${\alpha }_1>0$ and ${\mathsf{\Sigma }}_3<0$

  $$\begin{array}{c}\hfill {\phi }_{13}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_3}}{4{\alpha }_1}}\left(tanh\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (20)

  $$\begin{array}{c}\hfill {\phi }_{14}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_3}}{4{\alpha }_1}}\left(coth\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (21)
  Subcase 2.2. If ${\alpha }_1>0\mathrm{and}{\mathsf{\Sigma }}_3>0$

  $$\begin{array}{c}\hfill {\phi }_{15}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{{\mathsf{\Sigma }}_3}}{4{\alpha }_1}}\left(tan\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (22)

  $$\begin{array}{c}\hfill {\phi }_{16}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{{\mathsf{\Sigma }}_3}}{4{\alpha }_1}}\left(cot\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{4\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (23)
  Case-3: For ${\alpha }_4={\displaystyle \frac{{\alpha }_2{\mathsf{\Sigma }}_1}{8{\alpha }_1^2}},{\alpha }_5={\displaystyle \frac{{\mathsf{\Sigma }}_1^2}{64{\alpha }_1^3}}.$ Solving the algebraic system yields

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_0={\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},A_1={\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-\left({\Omega }^2+1\right)\right)}}},\omega ={\displaystyle \frac{1}{16}}\left(\left(8{\alpha }_3-{\displaystyle \frac{3{\alpha }_2^2}{{\alpha }_1}}\right){\zeta }^2-8{\kappa }^2\right).\hfill \end{array}\right.\end{array}$$
• The hyperbolic and trigonometric solutions
  Subcase 3.1. If ${\alpha }_1<0$ and ${\mathsf{\Sigma }}_3<0$

  $$\begin{array}{c}\hfill {\phi }_{17}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_3}}{2\sqrt{2}{\alpha }_1}}\left(\sec \mathrm{h}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\sqrt{2}\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (24)

  $$\begin{array}{c}\hfill {\phi }_{18}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{{\mathsf{\Sigma }}_3}}{2\sqrt{2}{\alpha }_1}}\left(\csc \mathrm{h}\left({\displaystyle \frac{\sqrt{{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\sqrt{2}\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (25)
  Subcase 3.2. If ${\alpha }_1>0\mathrm{a}\mathrm{n}\mathrm{d}{\mathsf{\Sigma }}_3<0$

  $$\begin{array}{c}\hfill {\phi }_{19}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_3}}{2\sqrt{2}{\alpha }_1}}\left(sec\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\sqrt{2}\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (26)

  $$\begin{array}{c}\hfill {\phi }_{20}(s,t)=\{{\displaystyle \frac{{\alpha }_1\zeta }{\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\left(-{\displaystyle \frac{{\alpha }_2}{4{\alpha }_1}}\pm {\displaystyle \frac{\sqrt{-{\mathsf{\Sigma }}_3}}{2\sqrt{2}{\alpha }_1}}\left(csc\left({\displaystyle \frac{\sqrt{-{\alpha }_1{\mathsf{\Sigma }}_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{2\sqrt{2}\alpha {\alpha }_1}}\right)\right)\right)+{\displaystyle \frac{{\alpha }_2\zeta }{4\sqrt{\mho }\sqrt{{\alpha }_1\left(-{\Omega }^2-1\right)}}}\}\times e^{i\chi }.\end{array}$$

  (27)
  Case-4: For ${\alpha }_2={\alpha }_4={\alpha }_5=0$ and ${\alpha }_3>0.$ Solving the algebraic system yields

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_0=0,A_1={\displaystyle \frac{\sqrt{{\alpha }_1}\zeta }{\sqrt{\mho }\sqrt{-{\Omega }^2-1}}},\omega ={\displaystyle \frac{1}{2}}\left({\alpha }_3{\zeta }^2-{\kappa }^2\right).\hfill \end{array}\right.\end{array}$$
• The exponential function solution

  $$\begin{array}{c}\hfill {\phi }_{21}(s,t)=\{{\displaystyle \frac{\sqrt{{\alpha }_1}\zeta }{\sqrt{\mho }\sqrt{-{\Omega }^2-1}}}\left({\displaystyle \frac{4{\alpha }_3\rho }{4{\rho }^2{e}^{\frac{\sqrt{{\alpha }_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }}-{\alpha }_1{\alpha }_3{e}^{-\frac{\sqrt{{\alpha }_3}\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }}}}\right)\}\times e^{i\chi }.\end{array}$$

  (28)

Remark 2. 

For all above twenty one solutions $\chi ={\displaystyle \frac{\Gamma (\beta +1)\left(\kappa s^{\alpha }+\omega t^{\alpha }\right)}{\alpha }}$.

3.2. Solutions via Modified Sardar Subequation Method

The solution to this method is given below:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\xi )=A_0+\sum _{h=1}^n{A}_h{\left(Q(\xi )\right)}^h.\end{array}$$

(29)

Moreover, $Q(\xi )$ fulfils the following first order ODE:

$$\begin{array}{c}\hfill [{\left(Q^{\prime }(\xi )\right]}^2=[g_2{Q}^4(\xi )+g_1{Q}^2(\xi )+g_0],\end{array}$$

(30)

where $g_i(i=0,1,2)$ are constants. Moreover, the exact solutions of Equation (30) can be observed by scrutinizing its distinct solution families for whole detail see the refrence [41]. For $n=1$, the Equation (29) simplifies to:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\xi )=A_0+A_{1}Q(\xi ).\end{array}$$

(31)

Substituting Equation (31) and its derivatives (derived from Equation (30)) into Equation (3), we gather coefficients of like powers of $Q(\xi )$. Setting these coefficients to zero and solving the resulting system using representative computation (via Mathematica) yields the subsequent solution sets.

$$\begin{array}{c}\hfill Set-1:A_0=0,\omega ={\displaystyle \frac{1}{2}}\left({\zeta }^2{g}_1-{\kappa }^2\right),A_1={\displaystyle \frac{i\zeta \sqrt{g_2}}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}}.\end{array}$$

For set-1, the following forms of solutions are established.

Case-1: If $g_0=0,g_1>0,\mathrm{and}{g}_2\ne 0$, we obtain the bright and explicit solitonic structure

$$\begin{array}{c}\hfill {\phi }_1(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)\sec \mathrm{h}\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}}\}.\end{array}$$

(32)

$$\begin{array}{c}\hfill {\phi }_2(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)\csc \mathrm{h}\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}}\}.\end{array}$$

(33)

Case-2: If for constants ${\beta }_1\mathrm{and}{\beta }_2.g_2=\pm 4{\beta }_1{\beta }_2,g_1>0,\mathrm{and}{g}_0=0,\mathrm{we}\mathrm{have}$

$$\begin{array}{c}\hfill {\phi }_3(s,t)=\{{\displaystyle \frac{8i{\beta }_1\sqrt{{\beta }_1{\beta }_2}\zeta \sqrt{g_1}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}\left(\left(4{\beta }_1^2-4{\beta }_1{\beta }_2\right)cosh\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)+\left(4{\beta }_1^2+4{\beta }_2{\beta }_1\right)sinh\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)\right)}}\}.\end{array}$$

(34)

Case-3: If $g_0={\displaystyle \frac{g_1^2}{4g_2}},g_1<0,\mathrm{and}{g}_2>0,$ we have dark, singular and combined wave structures

$$\begin{array}{c}\hfill {\phi }_4(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)tanh\left(\frac{\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)}{\sqrt{2}}\right)}{\sqrt{2\mho }\sqrt{{\Omega }^2+1}}}\}.\end{array}$$

(35)

$$\begin{array}{c}\hfill {\phi }_5(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)coth\left(\frac{\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)}{\sqrt{2}}\right)}{\sqrt{2\mho }\sqrt{{\Omega }^2+1}}}\}.\\ \hfill {\phi }_6(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}\left(tanh\left(\sqrt{2}\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)-i\sec \mathrm{h}\left(\sqrt{2}\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)\right)}{\sqrt{2\mho }\sqrt{{\Omega }^2+1}}}\}\end{array}$$

(36)

$$\begin{array}{c}\hfill \times exp\left({\displaystyle \frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }}\right).\end{array}$$

(37)

$$\begin{array}{c}\hfill {\phi }_7(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}\left(coth\left(\frac{\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)}{2\sqrt{2}}\right)+tanh\left(\frac{\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)}{2\sqrt{2}}\right)\right)}{2\sqrt{2\mho }\sqrt{{\Omega }^2+1}}}\}\\ \hfill \times exp\left({\displaystyle \frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }}\right).\end{array}$$

(38)

Case-4: If $g_0=0,g_1<0,\mathrm{and}{g}_2\ne 0,\mathrm{we}\mathrm{obtain}\mathrm{explicit}\mathrm{trigonometric}\mathrm{solutions}$

$$\begin{array}{c}\hfill {\phi }_8(s,t)=\{-{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)sec\left(\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}}\}.\end{array}$$

(39)

$$\begin{array}{c}\hfill {\phi }_9(s,t)=\{-{\displaystyle \frac{i\zeta \sqrt{-\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)csc\left(\sqrt{-g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}}\}.\end{array}$$

(40)

Case-5: If $g_0={\displaystyle \frac{g_1^2}{4g_2}},g_1>0,\mathrm{and}{g}_2>0,\mathrm{we}\mathrm{obtain}\mathrm{mixed}\mathrm{periodic}\mathrm{solutions}$

$$\begin{array}{c}\hfill {\phi }_{10}(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{\frac{g_1}{g_2}}\sqrt{g_2}\left(tan\left(\sqrt{2}\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)+sec\left(\sqrt{2}\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)\right)}{\sqrt{2\mho }\sqrt{{\Omega }^2+1}}}\}\\ \hfill \times exp\left({\displaystyle \frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }}\right).\end{array}$$

(41)

$$\begin{array}{c}\hfill {\phi }_{11}(s,t)=\{{\displaystyle \frac{i\zeta \sqrt{\frac{g_1}{g_2}}\sqrt{g_2}exp\left(\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)cos\left(\sqrt{2}\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)}{\sqrt{2\mho }\sqrt{{\Omega }^2+1}\left(sin\left(\sqrt{2}\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)+1\right)}}\}.\end{array}$$

(42)

Case-6: If $g_0=0\mathrm{and}{g}_1>0.$

$$\begin{array}{c}\hfill {\phi }_{12}(s,t)=\{{\displaystyle \frac{4i\zeta g_1\sqrt{g_2}exp\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)+\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}\left(exp\left(2\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)-4g_1{g}_2\right)}}\}.\end{array}$$

(43)

$$\begin{array}{c}\hfill {\phi }_{13}(s,t)=\{{\displaystyle \frac{4i\zeta g_1\sqrt{g_2}exp\left(\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)+\frac{i\Gamma (\beta +1)\left(\frac{1}{2}{t}^{\alpha }\left({\zeta }^2{g}_1-{\kappa }^2\right)+\kappa s^{\alpha }\right)}{\alpha }\right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}\left(1-4g_1{g}_{2}exp\left(2\sqrt{g_1}\left(\frac{\Gamma (\beta +1)\left(\zeta s^{\alpha }-\zeta \kappa t^{\alpha }\right)}{\alpha }+\vartheta \right)\right)\right)}}\}.\end{array}$$

(44)

3.3. Solutions via Generalized Arnous Method

The generalized Arnous method involves assuming a solution of the form

$$Q(\xi )={ϵ}_0+\sum _{k=1}^n{\displaystyle \frac{{ϵ}_k+{\sigma }_k{\mathsf{\Phi }}^{\prime }{(\xi )}^k}{\mathsf{\Phi }{(\xi )}^k}}.$$

(45)

For $n=1$, this method proposes a solution to Equation (3) in the following form

$$Q(\xi )={ϵ}_0+{\displaystyle \frac{{ϵ}_1+{\sigma }_1{\mathsf{\Phi }}^{\prime }(\xi )}{\mathsf{\Phi }(\xi )}}.$$

(46)

By substituting Equation (46) into Equation (3) along with its derivatives ${\left({\mathsf{\Phi }}^{\prime }(\xi )\right)}^2=({\mathsf{\Phi }}^2(\xi )-\rho ){log}^2\left(\delta \right))$, we receive a polynomial in term of ${\displaystyle \frac{1}{\mathsf{\Phi }(\xi )}}{\displaystyle \frac{{\mathsf{\Phi }}^{\prime }(\xi )}{\mathsf{\Phi }(\xi )}}$. Collecting and equating coefficients, A set of algebraic equations is obtained, yielding these solution sets:

$$\begin{array}{ccc}\hfill Set-1& :& {ϵ}_0=0,{ϵ}_1={\displaystyle \frac{\zeta \sqrt{\rho }log\left(\delta \right)}{\sqrt{\mho }\sqrt{{\Omega }^2+1}}},{\sigma }_1=0,\omega ={\displaystyle \frac{1}{2}}\left({\zeta }^2{log}^2\left(\delta \right)-{\kappa }^2\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Set-2& :& {ϵ}_0=0,{ϵ}_1=0,{\sigma }_1={\displaystyle \frac{-i\zeta }{\sqrt{\mho }\sqrt{{\Omega }^2+1}}},\omega =-{\zeta }^2{log}^2\left(\delta \right)-{\displaystyle \frac{{\kappa }^2}{2}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Set-3& :& {ϵ}_0=0,{ϵ}_1=-{\displaystyle \frac{\zeta \sqrt{\rho }log\left(\delta \right)}{2\sqrt{\mho }\sqrt{{\Omega }^2+1}}},{\sigma }_1={\displaystyle \frac{-i\zeta }{2\sqrt{\mho }\sqrt{{\Omega }^2+1}}},\omega ={\displaystyle \frac{1}{4}}\left(-{\zeta }^2{log}^2\left(\delta \right)-2{\kappa }^2\right).\hfill \end{array}$$

At $\delta =e,\rho =4A^2.$ We obtain the diverse form of solutions.

According to set-1. We establish solitary wave solution in the following form

$$\begin{array}{c}\hfill {\phi }_1(s,t)={\displaystyle \frac{\sqrt{A^2}\zeta exp\left(\frac{i\Gamma (\beta +1)\left(\kappa s^{\alpha }+\frac{1}{2}\left({\zeta }^2-{\kappa }^2\right)t^{\alpha }\right)}{\alpha }\right)\sec \mathrm{h}\left(\frac{\zeta \Gamma (\beta +1)s^{\alpha }}{\alpha }-\frac{\zeta \kappa \Gamma (\beta +1)t^{\alpha }}{\alpha }\right)}{\sqrt{\mho }A\sqrt{{\Omega }^2+1}}}.\end{array}$$

(47)

According to set-2. We obtain hyperbolic solution in this form

$$\begin{array}{c}\hfill {\phi }_2(s,t)={\displaystyle \frac{\zeta exp\left(\frac{i\Gamma (\beta +1)\left(\kappa s^{\alpha }+\left(-{\zeta }^2-\frac{{\kappa }^2}{2}\right)t^{\alpha }\right)}{\alpha }\right)tanh\left(\frac{\zeta \Gamma (\beta +1)s^{\alpha }}{\alpha }-\frac{\zeta \kappa \Gamma (\beta +1)t^{\alpha }}{\alpha }\right)}{\sqrt{\mho }\sqrt{-{\Omega }^2-1}}}.\end{array}$$

(48)

According to set-3. We obtain combined hyperbolic solution in this form

$$\begin{array}{c}{\phi }_3(s,t)=exp\left({\displaystyle \frac{i\Gamma (\beta +1)\left(\kappa s^{\alpha }+\frac{1}{4}\left(-{\zeta }^2-2{\kappa }^2\right)t^{\alpha }\right)}{\alpha }}\right)\times \hfill \\ \left({\displaystyle \frac{\zeta \sqrt{{\Omega }^2+1}tanh\left(\frac{\zeta \Gamma (\beta +1)s^{\alpha }}{\alpha }-\frac{\zeta \kappa \Gamma (\beta +1)t^{\alpha }}{\alpha }\right)}{2\sqrt{\mho }\sqrt{-{\Omega }^4-2{\Omega }^2-1}}}-{\displaystyle \frac{\sqrt{A^2}\zeta \sqrt{-{\Omega }^2-1}\sec \mathrm{h}\left(\frac{\zeta \Gamma (\beta +1)s^{\alpha }}{\alpha }-\frac{\zeta \kappa \Gamma (\beta +1)t^{\alpha }}{\alpha }\right)}{2\sqrt{\mho }A\sqrt{-{\Omega }^4-2{\Omega }^2-1}}}\right).\hfill \end{array}$$

(49)

Remark 3. 

Using the same procedure as for $\phi (s,t)$, we can obtain the solutions for the three approaches mentioned above: $\gamma (s,t)=\Omega \phi (s,t)$.

4. Modulation Instability

The M-truncated fractional derivative, a regularized operator with a truncated Mittag-Leffler kernel, governs the non-local dynamics of the complex coupled system; it converges to the classical derivative as order $\alpha \to 1$, ensuring consistency. The system is well-posed under appropriate initial and boundary data, with existence and uniqueness established via fixed-point arguments under constraints on nonlinearity and fractional order. The modulation instability dispersion relation remains valid under this derivative, aligning with the classical limit and imposing parameter constraints such as weak nonlinearity and a proximity to $\alpha =1$. The validity of all solutions is contingent on specific parameter constraints, which include conditions on the wave number and a proximity to the classical order to maintain both physical meaning and mathematical integrity. Here, we use the concept of linear stability to analyze the modulation instability (MI) of the IOPM [47]. The perturbed solution of the IOPM can be found by

$$\begin{array}{c}\hfill \phi (s,t)=(\sqrt{\mu }+P(s,t))e^{i\mu t},\\ \hfill \gamma (s,t)=(\sqrt{\mu }+Q(s,t))e^{i\mu t},\end{array}$$

(50)

where $\mu$ is the steady state solution for Equation (1). By substituting Equation (50) with Equation (1) and then linearizing, we obtain

$$\begin{array}{c}\hfill \mu (8\mho -2)(P+P^{\ast })+4\mho \mu Q+P_{ss}+2i{P}_t=0,\end{array}$$

(51)

$$\begin{array}{c}\hfill \mu (8\mho -2)(Q+Q^{\ast })+4\mho \mu P+Q_{ss}+2i{Q}_t=0.\end{array}$$

(52)

In our analysis, we utilize the standard notation where complex conjugation is shown by an asterisk superscript (*). Regarding the governing Equations (51) and (52), we hypothesis solutions of the form.

$$\begin{array}{c}\hfill P(s,t)=f_1{e}^{i(ls-t\varpi )}+f_2{e}^{-i(ls-t\varpi )}.\end{array}$$

(53)

$$\begin{array}{c}\hfill Q(s,t)=f_1{e}^{i(ls-t\varpi )}+f_2{e}^{-i(ls-t\varpi )}.\end{array}$$

(54)

$$\begin{array}{c}\hfill P{(s,t)}^{\ast }=f_1{e}^{-i(ls-t\varpi )}+f_2{e}^{i(ls-t\varpi )}.\end{array}$$

(55)

$$\begin{array}{c}\hfill Q{(s,t)}^{\ast }=f_1{e}^{-i(ls-t\varpi )}+f_2{e}^{i(ls-t\varpi )}.\end{array}$$

(56)

In our perturbation analysis, we denote the normalized wave number and frequency by l and $\varpi$, respectively.

To examine the stability of Equation (51), we substitute Equations (53)–(56) into Equation (51) and separate the coefficients of the harmonic components $e^{i(lx-\varpi t)}$ and $e^{-i(lx-\varpi t)}$. This yields the dispersion relation:

$$l^4-24\mho \mu l^2+4\mu l^2-4{\varpi }^2=0.$$

(57)

Solving for the perturbation frequency $\varpi$, we ecquire

$$\begin{array}{c}\hfill \varpi ={\displaystyle \frac{1}{2}}l\sqrt{4(1-6\mho )\mu +l^2}.\end{array}$$

(58)

The stability of the steady-state solution depends critically on the nature of $\varpi$. Stable regime: If $\varpi$ has a non-zero real part, the solution remains stable under small perturbations. Unstable regime: If $\varpi$ is purely imaginary, perturbations grow exponentially, leading to instability. The condition for instability is thus given by:

$$\begin{array}{c}\hfill 4(1-6\mho )\mu +l^2<0.\end{array}$$

Finally, the modulation instability (MI) gain spectrum in Figure 1, represented by $G\left(\mu \right)$, can be derived as follows:

$$\begin{array}{c}\hfill G\left(\mu \right)=2Im\left(\varpi \right)=2Im\left({\displaystyle \frac{1}{2}}l\sqrt{4(1-6\mho )\mu +l^2}\right).\end{array}$$

(59)

Remark 4. 

Through a similar process, we can obtain insights into the qualitative behavior of Equation (52).

5. Discussion and Visual Illustration

This section presents the results of our suggested model and offers visual illustrations. It emphasizes the extraction of generalized and innovative solitonic wave structures, encompassing a variety of solutions. Dark solitons, while more difficult to control than conventional solitons, exhibit superior stability and resilience to losses. Conversely, bright solitons are distinguished by their greater intensity relative to the surrounding background, whereas singular solitons define a different category of solitary waves that display singularities, usually appearing as endless discontinuities. Notably, these unique solitons may relate to solitary waves with imaginary center points. Examining unique solitons is particularly important as they may elucidate the formation of rogue waves, characterized by abrupt, extreme crests. Economically singular solutions are interpreted as regularized discontinuities, representing extreme events such as market crashes, while the calibration is designed to ensure the model converges to the classical Black–Scholes framework in the limit of vanishing asset interactions. Similarly, periodical wave patterns represent waves characterized by perpetual, repetitive patterns, dictated through their wavelength and frequency. The primary characteristics of these solutions include the period, which is the duration required for completing the wave cycle and the frequency, indicating the number of cycles that occur every second. These solutions possess unique physical depictions, which we depict schematically by selecting suitable value for parameters. The findings provide motivation for further study in various scientific disciplines, especially in fluid dynamics. In this work, we have produced multiple solutions for solitary waves by utilizing three effective strategies. Furthermore, our solutions offer insights for additional exploration of higher-order NLPDEs. Illustration is crucial for precisely portraying nonlinear occurrences and links among factors in a collection of data. The accompanying 2D, 3D and contour plots in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. Periodic singular solitons (Figure 2 and Figure 10) with parameters ($\alpha =0.98,{\alpha }_1=-0.85,$ ${\alpha }_2=0.35$, ${\alpha }_3=2.33,\mho =2.77,\beta =0.37,\zeta =1.2,\kappa =1.4,\omega =1.3,\mathrm{and}$ $\Omega =0.70.$ For Figure 2; $g_1=0.77,\alpha =0.97,A=0.5,\beta =0.71,\zeta =3.1,g_2=0.8,$ $\kappa =4.1,$ $\omega =3.07,\Omega =0.3,\mathrm{and}\vartheta =0.2.$ For Figure 10) demonstrate repeating wave patterns, while bright solitons ((Figure 4 and Figure 12): $\alpha =0.98,{\alpha }_1=-0.85,{\alpha }_2=0.35,$ ${\alpha }_3=2.33$, $\mho =2.77,\beta =0.37,\zeta =1.2,\kappa =1.4,\omega =1.3,\mathrm{and}\Omega =0.70.$ For Figure 4; $\zeta =1.1,\alpha =0.97,A=0.2,\beta =0.71,\mho =0.5,\kappa =0.89,\omega =3.07,\mathrm{and}\Omega =0.3.$ For Figure 12) exhibit localized intensity peaks on continuous backgrounds. Dark solitons (Figure 6 and Figure 14): $g_1=-0.47,\alpha =0.96,\mho =0.5,\beta =0.91,\zeta =2.3,$ $g_2=1.5,$ $\kappa =1.2,\omega =0.77,\Omega =1.4,\mathrm{and}\vartheta =0.2.$ For Figure 6; $\zeta =1.2,\alpha =0.94,$ $\beta =0.51$, $\mho =0.6,\kappa =0.99,\omega =3.07,\mathrm{and}\Omega =0.3.$ For Figure 14) show wave depressions and hyperbolic solutions ((Figure 8 and Figure 16): $g_1=-0.57,\alpha =0.95,\mho =0.6,$ $\beta =0.81,$ $\zeta =1.1,g_2=1.4,\kappa =1.4,\omega =0.27,\Omega =1.3,\mathrm{and}\vartheta =0.2.$ for Figure 8; $\zeta =0.7,$ $\alpha =0.99,$ $A=0.3,\beta =0.61,\mho =0.9,\kappa =1.79,\omega =3.07,\mathrm{and}\Omega =0.5.$ For Figure 16) reveal characteristic nonlinear profiles. Complementary 2D plots (Figure 3, Figure 5, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17) (odd-numbered Figures) decompose each solution into real, imaginary and absolute components, enabling amplitude-phase analysis. Visually the solutions obtained, facilitating a clearer understanding of problem-solving approaches. It is easy to understand these visual images of complicated ideas and methods in nonlinear wave analysis. Therefore, adding fractional calculus to option-pricing models provides a more accurate representation of the dynamics of financial markets, improving risk-management techniques and facilitating better decision-making. By enhancing option value and market forecasting, Soliton technologies help financial institutions reduce risk and increase returns. In addition to finance, our work advances material science, biomedical engineering, signal processing, telecommunication and environmental sciences. For illustration, soliton behavior may be used to mimic the characteristics of complicated materials and improve signal integrity in telecommunications. Additionally, better medical imaging and diagnostics may be achieved by comprehending how waves propagate across biological tissues. By using these model challenges in many sectors, significant technical breakthroughs may be made, including better financial software, creative engineering solutions and improved diagnostic tools, which will spur innovation and success in a variety of fields.

Comparison

In this section, we provide a detailed comparison of our results with those previously reported in the literature. In 2024, Rehman et al. [48] investigated the same model and derived various soliton solutions including dark, bright, kink and singular types using the generalized Ricatti mapping method and the new Kudryashov method. Also, in 2024, Jahangir et al. [49] constructed multiple traveling wave solutions for the model and further conducted dynamical analyses. In the present study, we extend this work by deriving novel soliton solutions specifically Jacobi elliptic, hyperbolic and trigonometric forms via the application of three distinct analytical techniques: the Kumar-Malik method, the modified Sardar sub-equation method and the generalized Arnous method. Furthermore, our work includes a modulation instability (MI) analysis, which has not been previously addressed for this model. A review of existing literature confirms that Jacobi-type solutions and modulation instability remained unexplored, underscoring the novelty of our contributions. The results obtained here provide deeper insight into the Ivancevic option-pricing model (IOPM), offering physical interpretations of option price wave fluctuations and stable pulses. These findings have broad applicability across nonlinear sciences, particularly in economics, finance and option-pricing theory.

6. Concluding Remarks

This study effectively applied the aforementioned methods to secure diverse traveling wave solutions, including Jacobi elliptic, exponential, hyperbolic and singular periodic solutions. All analytical solutions reported in this study have been verified by direct substitution into the governing equations. These findings are significant for forecasting future option price wave functions and reveal a strong correlation between adaptive market potential, incident power and dispersion frequency coefficients. The obtained solutions hold broad implications for nonlinear sciences, particularly in economics, finance and option pricing, and suggest new methods for controlling long-term asset price fluctuations. The efficacy of the aforementioned methods is highlighted in solving nonlinear models in economics and beyond. The hyperbolic functions derived have many applications in fields such as physics and mathematics, including laminar jet profiles, gravitational potential calculations, the Schwarzschild metric in general relativity and special relativity’s treatment of rapidity and magnetic moment. Stability analysis of the studied problem has further underscored the reliability of these solutions. Visualizing the solutions through 3D, 2D and contour wave profiles, as shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, explain the physical behavior of the solutions with the choice of suitable parametric values. This extensive analysis establishes the proposed methods such that Kumar-Malik technique, the modified Sardar sub-equation approach and the modified Arnous approach as robust tools for preserving the essential physical features of real-world processes in option pricing. In summary, this study highlights a fractional nonlinear coupled wave system as a promising alternative to the Black–Scholes model. Numerical simulations reveal its heightened accuracy, exhibited by low error norms and strong convergence behavior when compared to benchmark solutions. Additionally, the model’s inherent flexibility permits for a more realistic representation of complex market dynamics. The above said three modern techniques effectively generate novel solitonic wave structures, while graphical representations clarify pulse propagation and parameter alignment. The flexibility of these methods, confirmed through comparative analysis of different fractional order values, marks a significant advancement in the field of option pricing. A future recommendation involves employing advanced numerical schemes, such as radial basis function (RBF)-based meshless methods, to validate these analytical solutions and further explore the model’s behavior under a wider range of market conditions.