Bifurcation Analysis and Solitons Dynamics of the Fractional Biswas–Arshed Equation via Analytical Method

Abstract

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering.

1. Introduction

Fractional calculus has fascinated many specialists due to its implications in uncovering deeper aspects of present reality. Although fractional derivatives were introduced nearly three centuries ago, they were not entirely a new concept [1]. Recent research on nonlinear fractional partial differential equations (NLFPDEs) has risen significantly, highlighting their diverse applications across various domains, including engineering, medical fields, controlled thermonuclear fusion, plasma physics, stochastic dynamical processes, acoustics, diffusive transport, chemical physics, solid state physics, electric networks, astrophysics, electromagnetic theory, fluid mechanics, geochemistry, fractional dynamics, manipulation theory, chemical kinematics, system identification, biogenetics, optical fibers, and many others [2,3,4,5]. The above-mentioned topics are closely related to engineering concepts such as response, diffusion, convection, and dispersion, and NLFPDEs can be applied to accurately solve them. Fractional calculus is the principal technique for differentiating and integrating averages of any order, enabling the construction and understanding of various physical features associated with continuous transitions from stationary to oscillatory phenomena. Due to the prevalence of fractional nonlinear differential equations (FNLDEs) throughout several engineering and scientific fields, numerous researchers focus their efforts on deriving exact or approximate solutions to these dynamic FNLDEs. Ahmed et al. [6] derive a rational solution, singular-periodic wave, periodic wave solution, dark, and collective dark–bright solutions of space-time NLFE for communication lines via the improved generalized Riccati equation technique.The single-wave solutions of the generalized Rosenau–Korteweg-de Vries (RKDv) regularized long-wave model were examined by Avazzadeh et al. [7]. Asjad et al. [8] examined nonlinear differential time-fractional equation to derive solutions like trigonometric functions, exponential functions, rational functions, and hyperbolic functions utilizing the EDA approach. Zaman et al. [9] investigated nonlinear fractional Allen Cahn (AC) and space-time Phi-4 equations via $\left({\displaystyle \frac{G^{\prime }}{G}}\right)$-expansion technique to obtain periodic soliton, singular kink, dynamical compaction, and single soliton-type solutions. Sadiya et al. [10] derived single solitons, kink type, bell shape, double solitons and multiple solitons types results by applying tanh-function technique to the Klein–Gordon and nonlinear time-fractional Sine-Gordon (SG) model. To analyze coupled approximate-long-water wave and space-time nonlinear fractional coupled systems, Zaman et al. [11] implemented the tanh-function method. To derive soliton solutions, such as bell-shape and kink-type solutions, use the Whitham–Broer–Kaup equation. Arefin et al. [12] utilized the Riemann–Liouville fractional derivative along with the two-variable $({\displaystyle \frac{G^{\prime }}{G}},{\displaystyle \frac{1}{G}})$-expansion technique to obtain exact solutions for the space-time NLF (2 + 1)-dimensional diffusive long-wave equation and the approximated long water-wave equation. During the application of the fractional differential sub-equation technique. RASHED et al. [13] examined the space-time NLF differential modified Benjamin Bona-Mahony model. Furthermore, two-variable $({\displaystyle \frac{G^{\prime }}{G}},{\displaystyle \frac{1}{G}})$-expansion technique [14,15], Sine–Gordon expansion method [16], $\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)$-expansion technique [17], modified extended tanh function approach [18], sine cosine and sinh cosh method [19], a novel modified extended direct algebraic method [20], extended generalized Riccati equation mapping method [21], Paul–Painlevé method [22], Laplace Pade–Caputo-fractional differential transform technique [23], generalized Kudryashov technique [24], explicit finite-difference technique [25], the Yang decomposition technique and homotopy perturbation technique [26], optimized decomposition technique [27], Chelyshkov wavelets method [28], differential inequality technique and integral transformation method [29], hybrid local fractional approach [30], natural decomposition technique [31], Mohand variational iteration transform technique and novel approximate analytical technique [32], iterative method [33], fractional new analytic technique [34], F-expansion approach [35], Elzaki transform technique [36], Adomia -decomposition and Jafari transform technique [37], Laplace residual power series approach [38], Bernstein spectral approach [39] are adopted by different researchers to solve NFFDEs. Here, our concern is to solve the Biswas arshed (BA) equation. Biswas and Arshed [40] introduced an innovative concept aimed at reducing issues by attaining an optimal equilibrium between dispersion and nonlinearity to preserve solitons in metamaterials, crystal fibers, optical fibers, and photonic crystal fibers, possibly leading to reduced group velocity dispersion and an absence of nonlinearity. In this instance, they recovered the combo, bright, and singular solitons to demonstrate that solitons still exist for BA equation. Many researchers have recently provided an explanation of the BA equation. Alhojilan et al. [41] find stochastic bright solutions, stochastic dark solutions, stochastic singular solutions, stochastic rational solution, stochastic singular-bright solutions, stochastic periodic solutions, and stochastic bright-dark solutions by investigating the Biswas–Arshed equation via modified extended mapping technique. Cinar et al. [42] used an extended sine cosine approach and extended sinh-cosh approach and obtain singular solitons solutions, dark solitons solutions, and singular-periodic soliton solutions from Biswas Arshed equation. Zhao Li et al. [43] obtain optical and traveling wave solitons solutions by investigating the BA equation including beta-derivative via Li’s three-step technique. Ghazala et al. [44] formed trigonometric function solutions, rational function solutions and hyperbolic function solutions by investigating BA model through the improved auxiliary equation method. Akbulut and Rayhanul [45] applied improved extended auxiliary equation and improved F-expansion technique to BA equation, including the $\beta$-derivative to gain rational function solutions, trigonometric function solutions, and hyperbolic function solutions. Sachin and Monika [46] investigated BA equation via a generalized exponential-rational function method and kudryashov’s technique to extract solitons solutions.

The governing model has the following form:

$$i{j}_t+a_1{j}_{xx}+a_2{j}_{xt}+i(b_1{j}_{xxx}+b_2{j}_{xxt})={i\left[\delta \right(\left|j\right|}^2{j)}_x+{\mu \left(\right|j|}^2{)}_{x}j+{\lambda \left|j\right|}^2{j}_x],$$

(1)

where $j(x,t)$ rely on x and t characterizes soliton profile, x represents the distance and t represents the time variables. $a_1$ represents the group velocity dispersion (GVD), while $a_2$, $b_1$ are the spatio-temporal dispersion coefficient and $b_2$ for the third-order dispersion coefficient. The nonlinearity is defined by the third component in the equation on the right-hand side, where $\delta$ denotes the influence of self-steepening, while $\lambda$ and $\mu$ indicate the nonlinear dispersion effect. Another powerful technique has been offered by Kudryashov in his research [47,48], the simplest equation technique is the name given to this methodology. Recently, Bilige et al. [49,50] modified this technique which is named the modified simplest equation technique. Subsequently, numerous researchers [51,52] applied this technique to gain novel exact solitons solutions for NLPDEs. This study uses extended-simplest equation technique to find new trigonometric, hyperbolic, and rational solutions for the BA equation and explains the resulting solutions, which include the $\beta$-derivative and M-truncated by their graphical representations. In this paper, We conduct a comprehensive comparison of the $\beta$ and M-truncated fractional derivatives applied to the Biswas–Arshed equation utilizing the Extended Simplest Equation technique. Unlike earlier research, which focused primarily on individual fractional derivatives or conventional analytical approaches, ours uniquely mixes beta and M-truncated derivatives within the same framework. This makes a direct comparative study, which the literature has not yet extensively investigated, feasible. Furthermore, we derive a more general and effective solution method, thereby generating a class of soliton solutions. This study offers innovative analytical perspectives that surpass prior research and elucidates the progression of fractional derivatives in nonlinear equations. This facilitates future progress in fractional calculus and its applications in complex physical systems.

This paper is organized as follows: Preliminaries are outlined in Section 2. The proposed approaches are summarized in Section 3. The governing models are outlined in Section 4. Section 5 discusses the application of the proposed methodologies to the governing system for extracting precise solutions. Section 6 presents graphical representations of several findings. Bifurcation analysis are described in Section 7. Section 8 contains the presentation of the conclusion and future work.

2. Preliminaries

2.1. $\beta$-Derivative

Definition 1 

(Atangana et al. [53]). Let $u\left(z\right)$ be a function that is defined for any non-negative z. Therefore, the β-derivative of $u\left(z\right)$ is as follows:

$$T^{\beta }\left(u\left(z\right)\right)={\displaystyle \frac{d^{\beta }u\left(z\right)}{d{z}^{\beta }}}={\displaystyle \underset{\xi \to 0}{lim}}{\displaystyle \frac{u\left(z+\xi {\left(z+\frac{1}{\Gamma \left(\beta \right)}\right)}^{1-\beta }\right)-u\left(z\right)}{\xi }},1<\beta \le 1.$$

(2)

Theorem 1. 

Let $u\left(z\right)$ and $v\left(z\right)$ be β-differentiable functions for all $z>0$ and $\beta \in (0,1]$ (Martínez et al. [54]). Then,

$$T^{\beta }(ru\left(z\right)+sv\left(z\right))=r{T}^{\beta }\left(u\left(z\right)\right)+s{T}^{\beta }\left(v\left(z\right)\right),\forall r,s\in \mathbb{R}.$$

$$T^{\beta }\left(u\left(z\right)v\left(z\right)\right)=v\left(z\right)T^{\beta }\left(u\left(z\right)\right)+u\left(z\right)T^{\beta }\left(v\left(z\right)\right).$$

$$T^{\beta }\left({\displaystyle \frac{u\left(z\right)}{v\left(z\right)}}\right)={\displaystyle \frac{v\left(z\right)T^{\beta }\left(u\left(z\right)\right)-u\left(z\right)T^{\beta }\left(v\left(z\right)\right)}{v^2\left(z\right)}}.$$

$$T^{\beta }\left(u\left(z\right)\right)={\left(z+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\right)}^{1-\beta }{\displaystyle \frac{du\left(z\right)}{dz}}.$$

2.2. M-Truncated Derivative

Definition 2. 

The Truncated-Mittag–Leffler function [55] is defined as follows:

$${}_i\mathbb{E}_{\beta }\left(g\left(z\right)\right)={\displaystyle \sum _{m=0}^i}{\displaystyle \frac{g{\left(z\right)}^m}{\Gamma (\beta m+1)}},$$

(3)

where $g\in \mathbb{C}$ and $\beta >0$

Definition 3. 

Let v: $[0,\infty )\to \left\}\right\}R$ be a function. The M-truncated derivative of v of order γ, where $\gamma \in (0,1)$ with respect to z, is defined as

$${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}v\left(z\right)={\displaystyle \underset{h\to 0}{lim}}{\displaystyle \frac{v(z+{}_i\mathbb{E}_{\beta }\left(h{z}^{-\beta }\right))-v\left(z\right)}{h}},\beta ,z>0,$$

(4)

where ${}_i\mathbb{E}_{\beta }(.)$ is the truncated Mittag–Leffler function.

Theorem 2. 

Let us assume that $0<\gamma \le 1,p,q\in \mathbb{R}$, $\beta >0$, ψ and φ be a γ-diferentiable at any $z>0$. Then,

$${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}(p\psi +q\phi )\left(z\right)=p{\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\psi \left(z\right)+q{\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\phi \left(z\right),p,q\in \mathbb{R}.$$

$${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left(\psi \phi \right)\left(z\right)=\psi \left(z\right){\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\phi \left(z\right)+\phi \left(z\right){\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\psi \left(z\right).$$

$${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left({\displaystyle \frac{\psi }{\phi }}\right)\left(z\right)={\displaystyle \frac{\psi \left(z\right){\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\phi \left(z\right)-\phi \left(z\right){\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\psi \left(z\right)}{\phi {\left(z\right)}^2}}.$$

If ψ is diferentiable, then

$${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left(\psi \right)\left(z\right)={\displaystyle \frac{z^{1-\gamma }}{\Gamma (\beta +1)}}{\displaystyle \frac{d\psi \left(z\right)}{dz}}.$$

3. Description of the Method

To explain this method, initially consider the following projective Riccati equation:

$$f^{\prime }\left(\chi \right)=-f\left(\chi \right)g\left(\chi \right),g^{\prime }\left(\chi \right)=-g^2\left(\chi \right)+\sigma f\left(\chi \right)-\rho ,\rho \ne 0$$

(5)

where $\rho$ and $\sigma$ are constants. Consider

$$f\left(\chi \right)={\displaystyle \frac{1}{\varphi \left(\chi \right)}},g\left(\chi \right)={\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}},\varphi \left(\chi \right)\ne 0$$

(6)

$\varphi \left(\chi \right)$ is the result of the following ODE of order two.

$$\sigma ={\varphi }^{″}\left(\chi \right)+\rho \varphi \left(\chi \right).$$

(7)

For this ODE, there are three different kinds of solutions.

(1) 

For $\rho <0$, hyperbolic function solution exist:

$$\varphi \left(\chi \right)=B_{1}cosh\left(\chi \sqrt{-\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+{\displaystyle \frac{\sigma }{\rho }}.$$

(8)

Due to this,

$${\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)}^2-{\Delta }_1\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)-{\displaystyle \frac{2\sigma }{\varphi \left(\chi \right)}}+\rho =0,$$

(9)

where, ${\Delta }_1=\rho B_1^2-\rho B_2^2-{\displaystyle \frac{{\sigma }^2}{\rho }}$

(2) 

For $\rho >0$, trigonometric function solution exist:

$$\varphi \left(\chi \right)=B_{1}cos\left(\chi \sqrt{\rho }\right)+B_{2}sin\left(\chi \sqrt{\rho }\right)+{\displaystyle \frac{\sigma }{\rho }}.$$

(10)

Due to this,

$${\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)}^2-{\Delta }_2\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)-{\displaystyle \frac{2\sigma }{\varphi \left(\chi \right)}}+\rho =0,$$

(11)

where, ${\Delta }_2=\rho B_1^2-\rho B_2^2-{\displaystyle \frac{{\sigma }^2}{\rho }}$

(3) 

For $\rho =0$, the solution is as follows:

$$\varphi \left(\chi \right)={\displaystyle \frac{\sigma {\chi }^2}{2}}+B_1\chi +B_2.$$

(12)

Due to this,

$${\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)}^2-{\Delta }_3\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)-{\displaystyle \frac{2\sigma }{\varphi \left(\chi \right)}}=0,$$

(13)

where, ${\Delta }_3=B_1^2-2\sigma B_2$.

Here, $B_1$ and $B_2$ are arbitrary constants.

Examine the nonlinear $(n+1)$-dimensional partial differential equation (PDE)

$$H(j,j_x,j_t,j_{xx},j_{xt},j_{tt},\dots )=0,$$

(14)

where F is a polynomial in $j(x,t)$ and its partial derivatives, containing the highest order derivatives and the nonlinear terms.

The key components of this methodology are outlined as follows:

• Stage (1): With the aid of a traveling wave theory.

  $$j(x,t)=J\left(\chi \right)e^{i\chi },\chi =x-{\displaystyle \frac{\tau }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta },\eta =-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ.$$

  $$j(x,t)=J\left(\chi \right)e^{i\eta },\chi =x-{\displaystyle \frac{\tau \Gamma (\beta +1)t^{\gamma }}{\gamma }},\eta =-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ.$$

  (15)

  The traveling wave theory transforms (15) the nonlinear PDE (14) into the nonlinear ODE as follows

  $$\mathbb{G}(j,j^{\prime },j^{″},\dots )=0,$$

  (16)

  where $\mathbb{G}$ is a polynomial in $j\left(\chi \right)$ and its total derivative with respect to $\chi$.
• Stage (2): Assume the following solution to Equation (16).

  $$j\left(\chi \right)={\displaystyle \sum _{i=0}^M}{\alpha }_i{\left[{\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right]}^i+{\displaystyle \sum _{j=0}^{M-1}}{\beta }_j{\left[{\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right]}^i{\displaystyle \frac{1}{\varphi \left(\chi \right)}},$$

  (17)

  where $\varphi \left(\chi \right)$ is the solution of second order ODE (7) and ${\alpha }_i$ and ${\beta }_j$ are arbitrary constants.
• Stage (3): By substitution (7), (9), (11), (13), and (17) to Equation (16), collect all terms with the same order of $\left({\displaystyle \frac{1}{{\varphi }^i\left(\chi \right)}}\right)$ and $\left({\displaystyle \frac{1}{{\varphi }^j\left(\chi \right)}}\right)\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)$ together and a set of algebraic equations results from setting them to zero. Find $\rho$, $\sigma$, ${\alpha }_i$, ${\beta }_i$$a_1$, $a_2$, $b_1$, $b_2$ by solving this system with the aid of Mathematica.

4. Governing Models

In this part, define the governing model equations including both $\beta$ derivatives and M-truncated derivatives.

4.1. Biswas Arshed Equation Including $\beta$-Derivative

Han T. et al. [56] defined BA equation including $\beta$-derivative as

$$i{\displaystyle \frac{{\partial }^{\beta }j}{\partial t^{\beta }}}+a_1{\displaystyle \frac{{\partial }^{2}j}{\partial x^2}}+a_2{\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\left({\displaystyle \frac{\partial j}{\partial x}}\right)-i\left(\delta {\displaystyle \frac{{\partial \left|j\right|}^{2}j}{\partial x}}+\mu j{\displaystyle \frac{{\partial \left|j\right|}^2}{\partial x}}+\lambda {\left|j\right|}^2{\displaystyle \frac{\partial j}{\partial x}}\right)$$

$$+i\left(b_1{\displaystyle \frac{{\partial }^{3}j}{\partial x^3}}+b_2{\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\left({\displaystyle \frac{{\partial }^{2}j}{\partial x^2}}\right)\right)=0,0<\beta \le 1,$$

(18)

where i represent the imaginary part, j is the function of x and t, ${\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}$ is the operator for $\beta$-derivative, $\beta$ represents fractional order, $a_1,a_2,b_1$, and $b_2$ are arbitrary constants, $\delta$, $\mu$, and $\lambda$ are nonzero parameters, the wave transformations listed below will be used:

$$j(x,t)=J\left(\chi \right)e^{i\eta },\chi =x-{\displaystyle \frac{\tau }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta },\eta =-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ,$$

(19)

where $\eta$$,\upsilon$, $\tau$, and $ϵ$ are non-zero parameters.

4.2. Biswas Arshed Equation Including M-Truncated Derivative

$$i{\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left(j\right)+a_1{\displaystyle \frac{{\partial }^{2}j}{\partial x^2}}+a_2{\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left({\displaystyle \frac{\partial j}{\partial x}}\right)-i\left(\delta {\displaystyle \frac{{\partial \left|j\right|}^{2}j}{\partial x}}+\mu j{\displaystyle \frac{{\partial \left|j\right|}^2}{\partial x}}+\lambda {\left|j\right|}^2{\displaystyle \frac{\partial j}{\partial x}}\right)$$

$$+i\left(b_1{\displaystyle \frac{{\partial }^{3}j}{\partial x^3}}+b_2{\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}\left({\displaystyle \frac{{\partial }^{2}j}{\partial x^2}}\right)\right)=0,0<\gamma \le 1,\beta >0,$$

(20)

where i represent the imaginary part, J is the function of x and t, ${\mathbb{D}}_{(M,z)}^{(\gamma ,\beta )}$ is operator for the $\gamma$-derivative, $\gamma$ represents fractional order, $a_1,a_2,b_1$, and $b_2$ are arbitrary constants, $\delta$, $\mu$, and $\lambda$ are nonzero parameters, the wave transformations listed below will be used:

$$j(x,t)=J\left(\chi \right)e^{i\eta },\chi =x-{\displaystyle \frac{\tau \Gamma (\beta +1)t^{\gamma }}{\gamma }},\eta =-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ,$$

(21)

where $\eta$, $\upsilon$, $\tau$, and $ϵ$ are non-zero parameters. In this study, we considered fractional operators defined on the whole memory interval [0, T]. However, in many real-world systems, distant past effects gradually diminish, and it is more realistic to restrict the fractional operator to a finite memory horizon, often referred to as the “short-memory principle.” In such cases, the fractional kernel is truncated so that only the most recent history contributes to the system’s dynamics. This concept has been shown to provide accurate models for viscoelastic and structural materials with fading memory effects [57,58]. Extending the Biswas–Arshed model to incorporate short-memory operators would therefore be an interesting future direction, offering a balance between computational efficiency and physical realism. The parameters $\beta$ and $\gamma$ represent the fractional orders in the $\beta$-derivative and the M-truncated derivative, respectively. Physically, these parameters quantify the extent of memory and hereditary effects in the system. When $\beta =1$ (or $\gamma =1$), the model reduces to the classical integer-order form without memory. For $0<\beta ,\gamma <1$, the model incorporates anomalous diffusion and long-range temporal correlations. Smaller values correspond to stronger memory effects, where past states significantly influence present dynamics, while larger values closer to unity indicate weaker memory and near-classical behavior. In the case of the M-truncated operator, $\gamma$ also introduces a tempered Mittag–Leffler kernel that effectively restricts the memory influence, making it particularly suitable for systems with fading memory such as viscoelastic materials oranomalous transport with cutoff effects.

5. Applications

This section employs the extended simplest equation method to resolve the Biswas–Arshed equation, including both $\beta$ and M-truncated derivatives. By substituting the wave transformations specified in (19) and (21) into (18) and (20), the resulting nonlinear ordinary differential equations are derived by separating the imaginary and real components, respectively.

$$(b_1-b_2\tau )J^{\left(3\right)}\left(\chi \right)+J^{\prime }\left(\chi \right)(d(a_2\tau -2a_1+2b_2\upsilon )+a_2\upsilon +d^2(b_2\tau -3b_1)-\tau )$$

$$-{\left(J\left(\chi \right)\right)}^2(\eta +3\delta +2\mu )J^{\prime }\left(\chi \right)=0.$$

(22)

$$J^{″}\left(\chi \right)(a_1-a_2\tau -2b_{2}d\tau +3b_{1}d-b_2\upsilon )-{\left(J\left(\chi \right)\right)}^{3}d(\eta +\delta )$$

$$+J\left(\chi \right)(\upsilon (a_{2}d+b_2{d}^2-1)-a_1{d}^2-b_1{d}^3)=0.$$

(23)

The following are the constraint conditions that can be found from Equation (22)

$$\tau ={\displaystyle \frac{b_1}{b_2}},\delta =-{\displaystyle \frac{1}{3}}(\eta +2\mu ),\upsilon =-{\displaystyle \frac{a_2{b}_{1}d-2a_1{b}_{2}d-2b_1{b}_2{d}^2-b_1}{b_2(a_2+2b_{2}d)}}.$$

(24)

Inserting Equation (24) into Equation (23)

$$d_1{J}^{″}\left(\chi \right)+d_{2}J\left(\chi \right)+d_3{\left(J\left(\chi \right)\right)}^3=0,$$

(25)

where

$$\begin{array}{ccc}\hfill d_1& =& -{\displaystyle \frac{a_2{b}_1}{b_2}}+a_1+b_{1}d+d_4,\hfill \\ \hfill d_2& =& -{\displaystyle \frac{a_2{d}_{4}d}{b_2}}-a_1{d}^2+{\displaystyle \frac{d_4}{b^2}}-b_1{d}^3-d_4{d}^2,\hfill \\ \hfill d_3& =& -\lambda d+{\displaystyle \frac{1}{3}}d(\lambda +2\mu ),\hfill \\ \hfill d_4& =& {\displaystyle \frac{a_2{b}_{1}d-2a_1{b}_{2}d-2b_1{b}_2{d}^2-b_1}{a_2+2b_{2}d}}.\hfill \end{array}$$

Equation (25) is used to determine the homogeneous balance number through the terms $J^{″}\left(\chi \right)$ and ${\left(J\left(\chi \right)\right)}^3$, we got M = 1 and the following formal solution from Equation (25).

$$J\left(\chi \right)={\displaystyle \frac{{\alpha }_1{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}+{\alpha }_0+{\displaystyle \frac{{\beta }_0}{\varphi \left(\chi \right)}}.$$

(26)

Type 1. For $\rho <0$.

Substitute Equation (26) with the linear OD Equation (7) and relation (9) into Equation (25), accumulate all the expression with the identical power of $\left({\displaystyle \frac{1}{{\varphi }^i\left(\chi \right)}}\right)$ and $\left({\displaystyle \frac{1}{{\varphi }^j\left(\chi \right)}}\right)\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)$ and put them equal to zero, the system of algebraic equations below is obtained.

Utilizing MATHEMATICA to resolve these nonlinear algebraic equations obtains the results given below:

Result:

$$\begin{array}{c}\left\{{\alpha }_0=0,{\alpha }_1=\pm {\displaystyle \frac{\sqrt{\frac{3}{2}}\sqrt{b_1}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right.,\hfill \\ {\beta }_0=\pm {\displaystyle \frac{\sqrt{\frac{3}{2}}\sqrt{b_1}\sqrt{{\Delta }_1}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}},\\ \hfill \left.a_1={\displaystyle \frac{b_1\left(a_2^2\rho +2a_2^2{d}^2-4a_{2}d+b_2\rho +2b_2{d}^2+2\right)}{b_2\left(a_2\rho +2a_2{d}^2-4d\right)}}\right\}\end{array}$$

(27)

From Equations (8), (15), (26) and (27), obtain the hyperbolic solution of Equation (1) as follows:

For $\beta$-derivative:

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(\sqrt{-\rho }{\vartheta }_1\left(\chi \right)+\sqrt{{\Delta }_1}{\vartheta }_1\left(\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ},$$

(28)

where

$${\vartheta }_1\left(\chi \right)={\displaystyle \frac{B_{1}sinh\left(\chi \sqrt{-\rho }\right)+B_{2}cosh\left(\chi \sqrt{-\rho }\right)}{B_{1}cosh\left(\chi \sqrt{-\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+\frac{\sigma }{\rho }}},$$

$${\vartheta }_2\left(\chi \right)={\displaystyle \frac{1}{B_{1}cosh\left(\chi \sqrt{-\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+\frac{\sigma }{\rho }}}.$$

In particular, if set $B_1=0,B_2\ne 0,\sigma =0$ in Equation (28), obtain singular wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(coth\left(\sqrt{-\rho }\chi \right)+csch\left(\sqrt{-\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ},$$

(29)

For M-truncated derivative:

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(\sqrt{-\rho }{\vartheta }_1\left(\chi \right)+\sqrt{{\Delta }_1}{\vartheta }_2\left(\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ},$$

(30)

where

$${\vartheta }_1\left(\chi \right)={\displaystyle \frac{B_{1}sinh\left(\chi \sqrt{-\rho }\right)+B_{2}cosh\left(\chi \sqrt{-\rho }\right)}{B_{1}cosh\left(\chi \sqrt{-\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+\frac{\sigma }{\rho }}},$$

$${\vartheta }_2\left(\chi \right)={\displaystyle \frac{1}{B_{1}cosh\left(\chi \sqrt{-\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+\frac{\sigma }{\rho }}}.$$

In particular, if set $B_1=0,B_2\ne 0,\sigma =0$ in Equation (30), obtain singular wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{\frac{3}{2}}\left(\sqrt{b_1}\sqrt{-\rho }\right)}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(coth\left(\sqrt{-\rho }\chi \right)+csch\left(\sqrt{-\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ}.$$

(31)

Type 2. For $\rho >0$.

Substitute Equation (26) with the linear OD Equation (7), and relation (11) into Equation (25), accumulate all the expression with the identical power of $\left({\displaystyle \frac{1}{{\varphi }^i\left(\chi \right)}}\right)$ and $\left({\displaystyle \frac{1}{{\varphi }^j\left(\chi \right)}}\right)\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)$ and put them equal to zero, the system of algebraic equations below is obtained.

By using Mathematica to solve these nonlinear algebraic equations and obtain results are shown below.

Result:

$$\begin{array}{c}\left\{{\alpha }_0=0,{\alpha }_1=\pm {\displaystyle \frac{\sqrt{\frac{3}{2}}\sqrt{b_1}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right.,\hfill \\ {\beta }_0=\pm {\displaystyle \frac{\sqrt{\frac{3}{2}}\sqrt{b_1}\sqrt{{\Delta }_2}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}},\\ \hfill \left.a_1={\displaystyle \frac{b_1\left(a_2^2\rho +2a_2^2{d}^2-4a_{2}d+b_2\rho +2b_2{d}^2+2\right)}{b_2\left(a_2\rho +2a_2{d}^2-4d\right)}}\right\}\end{array}$$

(32)

From Equations (10), (15), (26) and (32), obtain the trigonometric solution of Equation (1) as follows:

For $\beta$-derivative:

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(\sqrt{\rho }{\vartheta }_1\left(\chi \right)+\sqrt{{\Delta }_2}{\vartheta }_2\left(\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ},$$

(33)

where

$${\vartheta }_1\left(\chi \right)={\displaystyle \frac{-B_{1}sin\left(\chi \sqrt{\rho }\right)+B_{2}cos\left(\chi \sqrt{\rho }\right)}{B_{1}cos\left(\chi \sqrt{\rho }\right)+B_{2}sin\left(\chi \sqrt{\rho }\right)+\frac{\sigma }{\rho }}},$$

$${\vartheta }_2\left(\chi \right)={\displaystyle \frac{1}{B_{1}cos\left(\chi \sqrt{\rho }\right)+B_{2}sin\left(\chi \sqrt{\rho }\right)+\frac{\sigma }{\rho }}}.$$

In particular, if set $B_1=0,B_2\ne 0,\sigma =0$, Equation (33) and obtain periodic wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(cot\left(\sqrt{\rho }\chi \right)+\csc \left(\sqrt{\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ}.$$

(34)

In particular, if set $B_1\ne 0,B_2=0,\sigma =0,$ Equation (33) and obtain periodic wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(tan\left(\sqrt{\rho }\chi \right)-\sec \left(\sqrt{\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon }{\beta }}{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}+t\right)}^{\beta }+ϵ},$$

(35)

For M-truncated derivative:

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{b_1}\sqrt{\frac{3}{2}}}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(\sqrt{\rho }{\vartheta }_1\left(\chi \right)+\sqrt{{\Delta }_2}{\vartheta }_2\left(\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ},$$

(36)

where

$${\vartheta }_1\left(\chi \right)={\displaystyle \frac{-B_{1}sin\left(\chi \sqrt{\rho }\right)+B_{2}cos\left(\chi \sqrt{\rho }\right)}{B_{1}cosh\left(\chi \sqrt{\rho }\right)+B_{2}sinh\left(\chi \sqrt{-\rho }\right)+\frac{\sigma }{\rho }}},$$

$${\vartheta }_2\left(\chi \right)={\displaystyle \frac{1}{B_{1}cos\left(\chi \sqrt{\rho }\right)+B_{2}sin\left(\chi \sqrt{\rho }\right)+\frac{\sigma }{\rho }}}.$$

In particular, if set $B_1=0,B_2\ne 0,\sigma =0$, Equation (36), obtain periodic wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{\frac{3}{2}}\left(\sqrt{b_1}\sqrt{\rho }\right)}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(cot\left(\sqrt{\rho }\chi \right)+\csc \left(\sqrt{\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ}.$$

(37)

In particular, if set $B_1\ne 0,B_2=0,\sigma =0$, Equation (36) and obtain periodic wave solution

$$J(x,t)=\left({\displaystyle \frac{\pm \sqrt{\frac{3}{2}}\left(\sqrt{b_1}\sqrt{\rho }\right)}{\sqrt{2a_2{b}_2\lambda d^3-2a_2{b}_2{d}^3\mu +a_2{b}_2\rho \lambda d-a_2{b}_2\rho d\mu -4b_2\lambda d^2+4b_2{d}^2\mu }}}\right)$$

$$\left(tan\left(\sqrt{\rho }\chi \right)-\sec \left(\sqrt{\rho }\chi \right)\right)e^{-dx+{\displaystyle \frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }}+ϵ}.$$

(38)

Type 3. For $\rho =0$.

Substitute Equation (26) with the linear OD Equation (7), and relation (13) into Equation (25), accumulate all the expression with the identical power of $\left({\displaystyle \frac{1}{{\varphi }^i\left(\chi \right)}}\right)$ and $\left({\displaystyle \frac{1}{{\varphi }^j\left(\chi \right)}}\right)\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)$ and put them equal to zero, the system of algebraic equations below is obtained.

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)}^0& :& -{\displaystyle \frac{1}{3}}{\alpha }_{0}d\left(2{\alpha }_0^2(\lambda -\mu )+3a_{1}d+3b_1{d}^2\right)\hfill \\ & & +{\displaystyle \frac{{\alpha }_0\left(a_{2}d+b_2{d}^2-1\right)\left(b_1\left(-a_{2}d+2b_2{d}^2+1\right)+2a_1{b}_{2}d\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill \left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)& :& -d\left({\beta }_{0}d\left(a_1+b_{1}d\right)+2{\alpha }_0^2{\beta }_0(\lambda -\mu )+4{\alpha }_1^2{\alpha }_0\sigma (\lambda -\mu )\right)\hfill \\ & & +{\displaystyle \frac{{\beta }_0\left(a_{2}d+b_2{d}^2-1\right)\left(b_1\left(-a_{2}d+2b_2{d}^2+1\right)+2a_1{b}_{2}d\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill {\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)}^2& :& 3a_1{\beta }_0\sigma +d\left(3b_1{\beta }_0\sigma -2(\lambda -\mu )\left({\alpha }_1^2\left({\alpha }_0{\Delta }_3+2{\beta }_0\sigma \right)+{\alpha }_0{\beta }_0^2\right)\right)\hfill \\ & & -{\displaystyle \frac{3{\beta }_0\sigma \left(b_2\left(2a_1{b}_{2}d+b_1\left(2b_2{d}^2+1\right)\right)+a_2{b}_1{b}_{2}d+a_2^2{b}_1\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill {\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)}^3& :& {\displaystyle \frac{1}{3}}(-2){\beta }_0\left({\beta }_0^{2}d(\lambda -\mu )-3{\Delta }_3\left(a_1+d\left({\alpha }_1^2(\mu -\lambda )+b_1\right)\right)\right)\hfill \\ & & -{\displaystyle \frac{2{\beta }_0{\Delta }_3\left(b_2\left(2a_1{b}_{2}d+b_1\left(2b_2{d}^2+1\right)\right)+a_2{b}_1{b}_{2}d+a_2^2{b}_1\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill \left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)& :& {\alpha }_1(-d)\left(2{\alpha }_0^2(\lambda -\mu )+a_{1}d+b_1{d}^2\right)\hfill \\ & & +{\displaystyle \frac{{\alpha }_1\left(a_{2}d+b_2{d}^2-1\right)\left(b_1\left(-a_{2}d+2b_2{d}^2+1\right)+2a_1{b}_{2}d\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill {\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)}^2\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)& :& -{\displaystyle \frac{1}{3}}{\alpha }_1\left(d\left(4(\lambda -\mu )\left(3{\alpha }_0{\beta }_0+{\alpha }_1^2\sigma \right)-3b_1\sigma \right)-3a_1\sigma \right)\hfill \\ & & -{\displaystyle \frac{{\alpha }_1\sigma \left(b_2\left(2a_1{b}_{2}d+b_1\left(2b_2{d}^2+1\right)\right)+a_2{b}_1{b}_{2}d+a_2^2{b}_1\right)}{b_2\left(a_2+2b_{2}d\right)}}=0,\hfill \\ \hfill {\left({\displaystyle \frac{1}{\varphi \left(\chi \right)}}\right)}^3\left({\displaystyle \frac{{\varphi }^{\prime }\left(\chi \right)}{\varphi \left(\chi \right)}}\right)& :& {\displaystyle \frac{1}{3}}(-2){\alpha }_1\left(3{\beta }_0^{2}d(\lambda -\mu )-{\Delta }_3\left(3a_1+d\left({\alpha }_1^2(\mu -\lambda )+3b_1\right)\right)\right)\hfill \\ & & -{\displaystyle \frac{2{\alpha }_1{\Delta }_3\left(b_2\left(2a_1{b}_{2}d+b_1\left(2b_2{d}^2+1\right)\right)+a_2{b}_1{b}_{2}d+a_2^2{b}_1\right)}{b_2\left(a_2+2b_{2}d\right)}}=0.\hfill \end{array}$$

Utilizing Mathematica to resolve these nonlinear algebraic equations obtains the results given below.

Result:

$$\begin{array}{c}\left\{{\alpha }_0=0,{\alpha }_1={\displaystyle \frac{\pm \sqrt{b_1}\sqrt{3}}{2\sqrt{a_2{b}_2\lambda d^3-a_2{b}_2{d}^3\mu -2b_2\lambda d^2+2b_2{d}^2\mu }}},\right.\hfill \\ {\beta }_0={\displaystyle \frac{\pm \sqrt{3}\sqrt{b_1}\sqrt{{\Delta }_3}}{2\sqrt{a_2{b}_2\lambda d^3-a_2{b}_2{d}^3\mu -2b_2\lambda d^2+2b_2{d}^2\mu }}},\\ \hfill \left.a_1={\displaystyle \frac{b_1\left(a_2^2{d}^2-2a_{2}d+b_2{d}^2+1\right)}{b_{2}d\left(a_{2}d-2\right)}}\right\}\end{array}$$

(39)

From Equations (12), (15), (26) and (39), obtain the rational solution of Equation (1) as follows:

For $\beta$-derivative:

$$J(x,t)={\displaystyle \frac{\left(\pm \sqrt{b_1}\sqrt{3}\right)\left(\sqrt{{\Delta }_3}{\vartheta }_2\left(\chi \right)+{\vartheta }_1\left(\chi \right)\right)e^{-dx+\frac{\upsilon }{\beta }{\left(\frac{1}{\Gamma \left(\beta \right)}+t\right)}^{\beta }+ϵ}}{2\sqrt{a_2{b}_2\lambda d^3-a_2{b}_2{d}^3\mu -2b_2\lambda d^2+2b_2{d}^2\mu }}},$$

(40)

For M-truncated derivative:

$$J(x,t)={\displaystyle \frac{\pm \left(\sqrt{b_1}\sqrt{3}\right)\left(\sqrt{{\Delta }_3}{\vartheta }_2\left(\chi \right)+{\vartheta }_1\left(\chi \right)\right)e^{-dx+\frac{\upsilon \Gamma (\beta +1)t^{\gamma }}{\gamma }+ϵ}}{2\sqrt{a_2{b}_2\lambda d^3-a_2{b}_2{d}^3\mu -2b_2\lambda d^2+2b_2{d}^2\mu }}},$$

(41)

where

$${\vartheta }_1\left(\chi \right)={\displaystyle \frac{B_1+\chi \sigma }{B_1\chi +B_2+\frac{{\chi }^2\sigma }{2}}},$$

$${\vartheta }_2\left(\chi \right)={\displaystyle \frac{1}{B_1\chi +B_2+\frac{{\chi }^2\sigma }{2}}}.$$

Since the Biswas–Arshed equation is a time-space fractional model, appropriate initial and boundary conditions are necessary to ensure well-posedness. In the context of optical solitons, the initial condition at $t=0$ typically corresponds to a localized input pulse with finite amplitude. Moreover, the boundary condition is taken as follows:

$${\displaystyle \underset{\left|x\right|\to \infty }{lim}}J(x,t)=0,$$

meaning that the soliton amplitude vanishes at spatial infinity. These conditions reflect the physical reality of solitary waves that are localized in space and evolve from prescribed input profiles in time.

6. Physical Behavior of Solitons

This section contained the physical characteristics of the gained solitons with M-truncated and $\beta$ fractional derivative like kink, singular and periodic- singular types.

In Figure 1, the physical behavior of solution J(x,t) with the M-truncated fractional derivative shown in Equation (29), in Figure 1a 3-dim, with $d=0.1$, $\nu =-0.5$, $\delta =-1$, $\lambda =-0.2$, $\mu =-1.2, a_2=0.2, b_1=2, b_2=-0.5; \xi =0; \beta =0.5; \tau =0.5$, Figure 1b 2-dim with $t=0, 1, 2, 3$ and Figure 1c contour.

In Figure 2, the physical behavior of solution J(x,t) with the $\beta$ fractional derivative shown in Equation (31), in Figure 2a 3-dim, with $d=0.1, \nu =-0.5, \delta =-1, \lambda =-0.2$, $\mu =-1.2, a_2=0.2, b_1=2, b_2=-0.5;\xi =0; \beta =0.5; \tau =0.5; \alpha =1$, Figure 2b 2-dim with $t=0, 1, 2, 3$ and Figure 2c contour.

In Figure 3, the physical behavior of solution J(x,t) with the M-truncated fractional derivative shown in Equation (34), in Figure 3a 3-dim, with $d=0.1, \nu =-0.5, \delta =-1, \lambda =-0.2$, $\mu =-1.2, a_2=0.2, b_1=2, b_2=-0.5; \xi =0; \beta =0.5; \tau =0.5$, Figure 3b 2-dim with $t=0, 1, 2, 3$ and Figure 3c contour.

In Figure 4, the physical behavior of solution J(x,t) with the $\beta$ fractional derivative shown in Equation (37), in Figure 4a 3-dim, with $d=0.1, \nu =-0.5, \delta =-1, \lambda =-0.2, \mu =-1.2$, $a_2=0.2, b_1=2, b_2=-0.5; \xi =0; \beta =0.5; \tau =0.5; \alpha =1$, Figure 4b 2-dim with $t=0, 1, 2, 3$ and Figure 4c contour.

In Figure 5, the physical behavior of solution J(x,t) with the M-truncated fractional derivative shown in Equation (35), in Figure 5a 3-dim, with $d=0.1, \nu =-0.5, \delta =-1, \lambda =-0.2, \mu =-1.2$, $a_2=0.2, b_1=2, b_2=-0.5; \xi =0; \beta =0.5; \tau =0.5$, Figure 5b 2-dim with $t=0, 1, 2, 3$ and Figure 5c contour.

In Figure 6, the physical behavior of solution J(x,t) with the $\beta$ fractional derivative shown in Equation (38), in Figure 6a 3-dim, with $d=0.1, \nu =-0.5, \delta =-1, \lambda =-0.2, \mu =-1.2$, $a_2=0.2, b_1=2, b_2=-0.5; \xi =0; \beta =0.8; \tau =0.5$, Figure 6b 2-dim with $t=0, 1, 2, 3$ and Figure 6c contour.

In Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 same behavior of the soliton with M-truncated and $\beta$ fractional derivative.

To highlight the influence of fractional dynamics, we performed a parametric analysis of the obtained soliton solutions with respect to the fractional orders $\beta$ and $\gamma$. The results show that varying $\beta$ or $\gamma$ strongly affects the amplitude, width, and stability of the soliton profiles. For example, decreasing $\beta$ from $0.9$ to $0.5$ leads to broader and slower-decaying waveforms, reflecting enhanced memory effects. Conversely, values closer to unity recover sharper, more localized solitons resembling the classical case. A similar trend is observed for $\gamma$, where smaller values correspond to stronger fading memory and slower propagation dynamics. These findings confirm that fractional orders act as tunable parameters controlling the balance between dispersion and nonlinearity, thereby offering a richer description of wave evolution in complex media.

7. Bifurication Analysis

The bifurcation analysis of the time-fractional BA model is investigated in this section using a planar dynamical system. In order to simplify this analysis, the dynamical system in Equation (25) is assumed to be represented as follows:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dP}{d\chi }}& =Q,\hfill \\ \hfill {\displaystyle \frac{dQ}{d\chi }}& =-{\displaystyle \frac{d_2}{d_1}}P-{\displaystyle \frac{d_3}{d_1}}{P}^3,\hfill \end{array}\right.$$

(42)

where $P\left(\chi \right)=J\left(\chi \right)$ and $Q\left(\chi \right)=J^{\prime }\left(\chi \right)$. This system displays wave solutions of the time-fractional BA model alongside the established phase portraits in the $(P,Q)$-plane. The Hamiltonian canonical equations $P^{\prime }\left(\chi \right)={\displaystyle \frac{\partial H}{\partial Q}}$ and $Q^{\prime }\left(\chi \right)=-{\displaystyle \frac{\partial H}{\partial P}}$ are applied to the equivalent Hamiltonian function, as illustrated in Equation (25) or Equation (42), to derive the correlated Hamiltonian function.

$$H(P,Q)={\displaystyle \frac{d_1}{2}}{Q}^2+{\displaystyle \frac{d_2}{2}}{P}^2+{\displaystyle \frac{d_3}{4}}{P}^4.$$

(43)

To assess the stability of the solution, H must remain constant along trajectories ($dH/d\chi =0$). The system (42) contains two equilibrium points (EPs): $E{P}_1=(0,0)$ and $E{P}_2=(\pm \sqrt{-{\displaystyle \frac{d_2}{d_3}}},0)$, dependent upon the condition ${\displaystyle \frac{d_2}{d_3}}<0$. Here, $E{P}_2$ functions as both a saddle point and a center, whereas $E{P}_1$ serves as a center point, exhibiting a trajectory as illustrated in the Figure 7. We can write the characteristic equation of the Jacobian matrix as follows:

$${\eta }^2=-({\displaystyle \frac{d_2}{d_1}}+3{\displaystyle \frac{d_3}{d_1}}{P}^2).$$

(44)

The characteristic roots of the $E{P}_1$ are $\pm \sqrt{E}$, where $E=-{\displaystyle \frac{d_2}{d_1}}$ and $d_1\ne 0$, in order to assure its stability. If $E>0$, the eigenvalues are real and of the opposite sign, implying that the specified $E{P}_1$ is a saddle point. The $E{P}_1$ is a stable center or of elliptic type if $E<0$ and the eigenvalues are imaginary. Alternatively, the characteristic roots are $\pm i\sqrt{2E}$ for the stability of the point $(\pm \sqrt{-{\displaystyle \frac{d_2}{d_3}}},0)$, where ${\displaystyle \frac{d_2}{d_3}}<0$. If $E>0$, the eigenvalues are imaginary, signifying that the given $E{P}_2$ is a stable center. If $E<0$, the eigenvalues are real and of opposite signs, signifying that $E{P}_2$ represents a saddle point. The phase portraits in Figure 7 illustrate how the bifurcation parameter $E={\displaystyle \frac{d_2}{d_1}}$ governs the stability of equilibrium points in the BA model. For $E>0$, Figure 7 demonstrate that $E{P}_1$ acts as an unstable saddle point with divergent trajectories, while $E{P}_2$ functions as a stable center enclosed by periodic orbits. On the contrary, when $E<0$, Figure 7, the stability is reversed: $E{P}_1$ converts into a stable center defined by closed orbits, while $E{P}_2$ becomes an unstable saddle. The contour plots provide additional validation for these systems, illustrating energy level curves that correspond with saddle (hyperbolic) and center (elliptic) dynamics. For the Hamiltonian system Equation (43), we produce three-dimensional representations of the total surface energy, as illustrated in Figure 8. This research highlights the model’s complex dynamical structure, with implications for solitary and periodic wave phenomena.

8. Conclusions

This study presents new optical soliton solutions to the time-fractional Biswas–Arshed (BA) equation by employing the extended simplest equation approach in conjunction with both $\beta$-fractional and M-truncated derivatives. The effectively established a range of solution types, including kink, singular, and periodic-singular forms, by choosing random values for the free parameters by using the computational program called Mathematica. These solutions are important because they provide new ideas and instruments to address the complex problems posed by higher-order and dispersive nonlinear partial differential equations. The findings not only improve the theoretical knowledge of the investigated models, but they also have potential uses in clarifying the physical events driven by these equations. The computational method shown here is a strong and efficient mathematical instrument that provides a solid basis for investigating wave solutions in a broad spectrum of complicated nonlinear models across several spheres of nonlinear research. This paper also presents a bifurcation analysis of a nonlinear second-order differential equation by transforming it to a planar dynamical system and determining its Hamiltonian structure. The critical points were obtained analytically, and their stability was assessed using linearization and eigenvalue analysis. It was demonstrated that the nature of the equilibrium points, whether saddle or center, relies on the sign of a bifurcation parameter derived from the system’s coefficients. This parameter defines the boundary between qualitatively distinct solution behaviors in the phase space. These results demonstrate the importance of bifurcation theory in anticipating structural changes in nonlinear systems. Although the present study establishes exact soliton solutions and bifurcation analysis for the fractional Biswas–Arshed equation, several limitations should be acknowledged. First, the analysis is restricted to two specific fractional operators: namely the $\beta$-derivative and the M-truncated derivative, whereas other modern definitions such as the Caputo–Fabrizio or Atangana–Baleanu operators may yield different behaviors. Second, the approach is primarily analytical and does not include full-scale numerical simulations of the governing PDE, which would be necessary to validate the robustness of the solutions under perturbations. Third, the obtained soliton solutions are derived under certain parametric constraints, which may limit their direct applicability to arbitrary parameter regimes. Finally, only whole-memory operators were considered, while short-memory formulations—known to be important in fading-memory materials—remain an open direction for future research.