The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

Abstract

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering.

1. Introduction

In recent times, the study of soliton solutions has become a highly active area of research in nonlinear fiber optics. Soliton characteristics are principally described by the nonlinear Schrödinger equation and its various forms, including a form of singleton self-phase modulation (SPM) that arises from the refractive index changes in optical fiber. This phenomenon generally occurs under Kerr law nonlinearity and is characterized by a cubic nonlinear structure, which is modeled by the power law nonlinearity of media. The third form of SPM is associated with optical Gaussian profiles together with optical solitons governed by a nonlinear logarithmic law. Besides these three forms of single nonlinear terms, the other types of SPM are governed by saturable and exponential laws, which are rarely observed. More complex forms of SPM generally involve two or more nonlinear components that are applicable to different devices, such as LiNbO₃ crystals. Other refractive index structures include the quadratic–cubic (QC) form, the generalized QC form, the anti-cubic (AC) form, and the generalized AC form of nonlinearity, which are usually known as cubic–quantal nonlinearity. The SPM form is parabolic, and its theory is based on the dual power laws of SPM, quadratic–cubic, anti-cubic, and generalized AC types, also known as refractive index structures. In this study, we investigate the cubic-quantic-septic-nonic (CQS) nonlinear structure within the framework of a triple power law. This analysis is based on both chromatic dispersion (CD) and the refractive index in nonlinear structures. The SPM model is

$$i{V}_t+a{V}_{xx}+F\left({\left|V\right|}^2\right)V=0,$$

(1)

where the initial term takes from temporal evolution, where $i=\sqrt{-1}$, although $F$ appear from SPM here, $x$ describes the space variable and $t$ signifies the time-related variable, and $V(x,t)$ denotes the wave profile. Therefore, the cubic-quantic-septic-nonic equation [1] is

$$i{V}_t+a{V}_{xx}+(b_1{\left|V\right|}^2+b_2{\left|V\right|}^4+b_3{\left|V\right|}^6+b_4{\left|V\right|}^8)V=0.$$

(2)

In fluid dynamics, the Davey–Stewartson (DS) equation is an important model. This model was established by Davey and Stewartson to describe the evolution of a three-dimensional wave packet in water with finite depth. This equation is formed by a system of partial differential equations that combine a real field (mean flow) and a complex field (wave amplitude). The soliton solutions of this model can take various forms, including periodic, dark, bright, exponential, and singular solitons. The DS equation is a three-dimensional soliton equation. The DS equation is [2]

$$i{V}_t+{\displaystyle \frac{1}{2}}{\sigma }^2(V_{xx}+{\sigma }^2{V}_{yy})+\lambda {\left|V\right|}^{2}V-V{R}_x=0,$$

$$R_{xx}-{\sigma }^2{R}_{yy}-2\lambda {\left({\left|V\right|}^2\right)}_x=0.$$

(3)

Here, $\lambda$ is a real number. If $\sigma =1$, it is known as the DS-I equation, and if $\sigma =-1$, it is the DS-II equation. The cases of focusing and de-focusing are characterized by the parameter $\lambda$. Moreover, two well-known examples of integrable equations in two space dimensions that arise as higher-dimensional generalizations of the nonlinear Schrödinger equation are DS-I and DS-II. These equations play an active role in understanding turbulence for plasma waves and other intriguing physical phenomena. Basically, they were made in terms of water waves, but their utilization has expanded to several sectors of the scientific and engineering fields, including fluid mechanics and plasma physics.

Fractional Derivative: Fractional calculus, which involves the integration and differentiation of fractional orders, is a natural extension of classical calculus. The concept of a fractional operator in fractional calculus is derived from classical calculus principles. It was first mentioned by Leibniz and L’Hospital in their correspondence in 1695. Later, various physicists and mathematicians defined fractional derivatives from their own points of view. Some widely used definitions of fractional derivatives are the Lacroix derivative [3], Riemann–Liouville derivative [4], Jumarie modified Riemann–Liouville (RL) derivative [5], conformable derivative [6], Caputo derivative [7], and beta derivative [8].

The memory effect, where the current state of a material or phenomenon is influenced by its entire previous state rather than just the immediate past, known as the memory effect, can be analyzed using the Riemann–Liouville (RL) and Caputo derivatives, making them useful for modeling in physical science and engineering, such as viscoelastic materials, anomalous diffusion, signal processing, etc. However, these definitions of derivative exhibit shortcomings in adhering to the fundamental principles of derivatives; for instance, for a constant function, the RL definitions of fractional derivative are unable to yield zero and do not satisfy the chain rule; for the only zero domain, conformable derivative produces zero; and Caputo is restricted to highly smooth functions and does follow the chain rule. In contrast, the beta fractional derivative, presented in 2016 by Atangana et al. [9], addresses all these issues. The beta fractional derivative in modeling is notable for its ability to generalize the classical integer-order derivative while retaining important mathematical properties, such as the Leibnitz rule and the chain rule. This feature facilitates the application of familiar analytical and numerical methods with minimal modifications. Unlike intricate fractional derivatives such as the Riemann–Liouville or Caputo derivative, the beta derivative simplifies the mathematical structure of the model introducing fractional dynamics through fractional order parameters. Although the beta derivative lacks memory effects, it is particularly useful in models where the aim is to study the effects of non-integer differentiation without complicating the interpretation of physical mechanisms. Thus, the beta fractional derivative strikes a balance between complexity and analytical controllability, making it suitable for exploring fractional dynamics in systems with less emphasis on memory effects. Therefore, in this article, we utilized the beta derivative. The $\beta$ derivative [10] is defined as

$${}_0{}^A{D}_t^{\beta }\chi \left(t\right)=\underset{h\to 0}{\lim }{\displaystyle \frac{\chi (t+h{\left(t+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\right)}^{1-\beta })-\chi (t)}{h}},t\ge 0,0<\beta \le 1,$$

(4)

where $\chi \left(t\right)$ is a differentiable function, and $\beta$ is the fractional-order derivative. In the oscillating phenomenon, different types of physical models are created and explained using fractional calculus for gradual changes from relaxation. The wave solution of the nonlinear fractional differential equation (NFDE) is an important portion of the investigation of nonlinear physical explication. The fundamental properties that the beta derivative satisfies are provided in the following:

• ${}_0{}^A{D}_t^{\beta }C=0$.
• ${}_0{}^A{D}_t^{\beta }\left[c_1\chi \left(t\right)+c_2\mathcal{K}\left(t\right)\right]=c_1{}_0{}^A{D}_t^{\beta }\chi (t)+c_2{}_0{}^A{D}_t^{\beta }\mathcal{K}(t)$.
• ${}_0{}^A{D}_t^{\beta }\left\{\chi \left(t\right)\mathcal{K}\left(t\right)\right\}=\chi \left(t\right){}_0{}^A{D}_t^{\beta }\mathcal{K}\left(t\right)+\mathcal{K}\left(t\right){}_0{}^A{D}_t^{\beta }\chi (t)$.
• ${}_0{}^A{D}_t^{\beta }{\displaystyle \frac{\chi (t)}{\mathcal{K}(t)}}={\displaystyle \frac{\mathcal{K}(t){}_0{}^A{D}_t^{\beta }\chi \left(t\right)-\chi (t){}_0{}^A{D}_t^{\beta }\mathcal{K}(t)}{{\mathcal{K}}^2(t)}}$.
• ${}_0{}^A{D}_t^{\beta }\chi \left(\mathcal{K}\left(t\right)\right)={\displaystyle \frac{d\chi \left(\mathcal{K}\left(t\right)\right)}{d\mathcal{K}\left(t\right)}}{}_0{}^A{D}_t^{\beta }\mathcal{K}(t)$.

Beside these fundamental properties, the beta derivative satisfies all other theorems and properties associated with the integer-order derivative [9]. Moreover, the beta derivative has a relationship with the classical derivative, which ensures that the beta derivative behaves similarly to the classical derivative in all aspects, highlighting its broader applicability and consistency. In the following, we will establish this relationship.

Put $\mathsf{\delta }=h{\left(\mathrm{t}+{\displaystyle \frac{1}{\mathsf{\Gamma }\mathsf{\beta }}}\right)}^{1-\beta }$. Thus, $h={\mathsf{\delta }\left(\mathrm{t}+{\displaystyle \frac{1}{\mathsf{\Gamma }\mathsf{\beta }}}\right)}^{\beta -1}$. Therefore, as $h\to 0$ implies $\delta \to 0$. Hence, we obtain

$${}_0{}^A{D}_t^{\beta }\chi \left(t\right)={\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\mathsf{\beta }}}\right)}^{1-\beta }\underset{\delta \to 0}{\mathit{lim}}{\displaystyle \frac{\chi \left(t+\delta \right)-\chi \left(t\right)}{\delta }}={\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\mathsf{\beta }}}\right)}^{1-\beta }{\displaystyle \frac{d\chi \left(t\right)}{dt}}.$$

The exact solitary wave solutions play a fundamental role in searching nonlinear equations in various fields, such as applied mathematics, science, and engineering. They provide a strong interpretation of the behavior of the system. They also make a bridge between theory and applications. From the exact solutions of an equation, one can understand the effect of the parameters on the system, identify critical points, and predict qualitative changes. The properties, advantages, and limitations of the beta derivative and its applicability are discussed in detail in references [10,11]. An in-depth analysis and potential use in various fields are also discussed in these references.

Exact solutions assist mathematicians, scientists, and engineers to understand, predict, and manipulate nonlinear phenomena. Therefore, in recent years, several researchers have developed and improved some analytical and asymptotic methods for solving nonlinear fractional differential equations (NFDEs). Some widely used methods and strategies are: the modified auxiliary equation method [12], the improved Bernoulli sub-equation function approach [13], the local fractional Riccati differential equation method [14], the fractional reduced differential transform approach [15], the $(G^{\prime }/G)$-expansion scheme [16,17], the sine–Gordon expansion process [18,19], the Lie symmetry technique [20], the homotopy analysis method [21], the Hirota bilinear procedure [22], the Cole–Hopf transformation [23], the new extended direct algebraic method [24], the F-expansion method [25], the variational iteration procedure [26], the modified simple equation approach [27], the first integral technique [28], the Adomian polynomial scheme [29], the fractional sub-equation scheme [30], the modified Kudryashov approach [31], the exp-function technique [32], etc.

Also, there are several studies on fractional derivatives and their applications. Torvik and Bagley [33] investigated the applicability of fractional derivatives in real materials. Gu and Ou [34] observed chaos in a fractional duplication system. Du et al. [35] showed using the least square method that the results obtained from fractional models in memory-related phenomena across mechanics, biology, and psychology fit experimental data. Tarasov [36] discussed the nature of fractional derivatives with integer-order derivatives. Luchko [37] explained fractional derivatives and the fundamental theorem of fractional calculus. Jisha et al. [38] investigated the fractional-order KS equation using the rational function method, etc.

Therefore, the objective of this article is to investigate nonlinear time-fractional models in optical fibers, specifically focusing on the cubic-quintic-septic-nonic equation and the Davey–Stewartson equation. In this study, we extract soliton solutions to the stated equations using the extended generalized Kudryashov method and examine the impact of fractional-order derivatives on these solutions, providing new insights into the behavior and dynamics of nonlinear models in optical fibers.

2. The Extended Generalized Kudryashov Method

The extended generalized Kudryashov method uses generalized ansatz functions with adjustable parameters, allowing the method to investigate a wide range of classical and fractional-order nonlinear evolution equations. The originality of this method is that it is capable of generating various exact solutions, such as solitons, rational, and periodic solutions, making it particularly useful for unraveling complex problems in fields such as mathematical physics and fluid dynamics. The ability to solve complex nonlinear equations such as time-fractional cubic-quintic-septic-nonic and Davey–Stewartson equations demonstrates the potentiality of the method.

Let us consider the following generic nonlinear equation:

$$H(u,{u_t^{\alpha },u}_x,u_y,u_{tt}^{2\alpha },u_{xt}^{2\alpha },u_{xx},\dots )=0,$$

(5)

where $H$ is a polynomial of $u$ and its partial derivative, which includes the highest-order derivative and nonlinear terms.

First step: Equation (5) can be transformed from fractional-order differential equation to a nonlinear integer-order differential equation by means of the fractional wave transformation. We consider the subsequent fractional transformation

$$u\left(x,y,t\right)=V\left(\xi \right),\xi =k_{1}x+k_{2}y-{\displaystyle \frac{v}{\alpha }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\right)}^{\alpha },$$

(6)

where $k_1$, $k_2$, and $v$ are nonzero constants used to minimize Equation (5) to the following equation

$$G(V,V^{\prime },V″,V‴,\dots )=0,$$

(7)

where $G$ is a polynomial in $V\left(\xi \right)$ and its derivatives $V^{\prime }\left(\xi \right)$, $V″\left(\xi \right)$, and so on. Here, $V^{\prime }={\displaystyle \frac{dV}{d\xi }}$.

Second step: In agreement with the extended Kudryashov approach, we consider the solution of Equation (7) in the following form:

$$V\left(\xi \right)={\displaystyle \frac{{\sum }_{i=0}^N{a}_i{R}^i}{{\sum }_{j=0}^M{C}_j{R}^j}},$$

(8)

where $a_i(i=0,1,2,3,\dots ,N)$ and $C_j(j=0,1,2,\dots ,M)$ are constants to be examined, such that $a_N\ne 0$ and $C_M\ne 0$. For the extended Kudryashov approach, we consider the following supporting equation:

$$R^{\prime }\left(\xi \right)=\left[R^{p+1}\left(\xi \right)-R\left(\xi \right)\right]\ln l,0<l\ne 1.$$

(9)

The general solution of Equation (9) is

$$R\left(\xi \right)={\left[{\displaystyle \frac{1}{1-\ln l A \mathrm{e}\mathrm{x}\mathrm{p}\left(p\xi \right)}}\right]}^{1/p},$$

(10)

where $A$ is an integral constant, and $p$ is a positive integer.

Third step: $N$ and $M$ are to be determined by balancing the highest-order derivative with the highest power of the nonlinear term in Equation (7).

Fourth step: Substituting Solution (8) and Equation (9) in Equation (7), a polynomial in $R^{ij}(i,j=0,1,2\dots .)$ can be founded. Collecting and equalizing each term of resemble power to zero, a group of algebraic equations can be found by using Mathematica to define the values of ${\lambda }_i$, ${\lambda }_j$, and $v$. By exploiting these values, the result can be obtained.

3. Determination of Soliton Solutions

Since fractional models can better express the intricate behavior and dynamics of physical phenomena, in this article, we consider fractional-order models instead of integer-order models. They are especially useful to explain long-range interactions that are difficult to describe by integer-order models. Fractional models offer further flexibility and precision in fields such as mechanics, biology, finance, and control systems, where integer-order models may not be sufficient.

3.1. The Time-Fractional Cubic-Quintic-Septic-Nonic Equation

In this section, we formulate assorted soliton solutions to the time-fractional cubic-quantic-septic-nonic equation with the help of the extended generalized Kudryashov expansion approach. We assume the model of the form [1]

$$i{u}_t^{\alpha }+a{u}_{xx}+(b_1{\left|u\right|}^2+b_2{\left|u\right|}^4+b_3{\left|u\right|}^6+b_4{\left|u\right|}^8)u=0.$$

(11)

Here, it is to be noted that $bj(1\le j\le 4)$ stems from $u^j$ for (1 $\le j\le 4)$ nonlinearity. Even though $u$ and $u^2$ are substantialized for LiNbO₃ crystals, $u^3$ and $u^4$ are negligibly inconsiderable. This part includes these nonlinearities to discuss the similar type of NLSE and inquiry its integrability aspect for the initial time. It should be noted that the insignificant contribution must be set to zero to ensure integrability. This would result in consistency within mathematical and physical problems. Consider the complex structure of the wave solution:

$$u\left(x,t\right)=V\left(\xi \right)e^{i\varphi \left(x,t\right)},$$

(12)

with

$$\xi =k\left(x+{\displaystyle \frac{v}{\alpha }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\right)}^{\alpha }\right),$$

(13)

and

$$\varphi \left(x,t\right)=-kx+{\displaystyle \frac{w}{\alpha }}{(t+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}})}^{\alpha }+\theta .$$

(14)

Here, $V\left(\xi \right)$ is the amplitude component, where $\xi$ is the wave variable, and $v$ is the soliton speed, and $\varphi \left(x,t\right)$ arises from the phase component, whereas $\theta$ is the phase constant, $w$ is the angular frequency, and $k$ is the wave number [1].

By substituting Equation (12) into Equation (11), it is transformed into a complex equation, where the real and imaginary parts give us the following simplest equation:

$$a{k}^{2}V″-V\left(a{k}^2+w\right)+b_4{V}^9+b_3{A}^7+b_2{V}^5+b_1{V}^3=0,$$

(15)

and

$$-k{V}^{\prime }\left(2ak+v\right)=0.$$

(16)

From Equation (16), is the following is obtained:

$$v=-2ak.$$

(17)

The balance number for Equation (15) is $1/4$. Thus, in order to simplify Equation (15), we use the next transformation:

$$V(\xi )={U\left(\xi \right)}^{{\displaystyle \frac{1}{4}}}.$$

(18)

Now, using the transformation (18), Equation (15) is converted to the following:

$$4a{k}^{2}UU″-3a{k}^{2}U{U^{\prime }}^2-16U^2\left(a{k}^2+w\right)+16b_1{U}^{{\displaystyle \frac{5}{2}}}+16b_3{U}^{{\displaystyle \frac{7}{2}}}+16b_4{U}^4+16b_2{U}^3=0.$$

(19)

Placing $b_1=b_3=0$ into Equation (19), we obtain

$$4a{k}^{2}UU″-3a{k}^{2}U{U^{\prime }}^2-16U^2\left(a{k}^2+w\right)+16b_4{U}^4+16b_2{U}^3=0.$$

(20)

Since in Equation (20), $b_1$ and $b_3$ are free from all term-bearing functional exponents, we can set $b_1=b_3=0$ simply for Equation (20) to be rendered integrable. Only $b_2$ and $b_4$ hold up to permit the integrability of Equation (20). Equation (20) and the main equation are equivalent to study with simply two non-zero terms addressing $b_2$ and $b_4$. It implies that the NLSE is integrable with a cubic–quintic nonlinear form.

Now, by balancing $UU″$ and $U^4$ appearing in Equation (20), we obtain

$$2\left(N-M\right)+2=4\left(N-M\right)\Rightarrow N=M+1.$$

Here $M$ is a free parameter; accordingly setting $M=1$, we obtain $N=2$. Thus, from (8), the solution structure of Equation (20) can be presented as follows:

$$U(\xi )={\displaystyle \frac{a_0+a_{1}R\left(\xi \right)+a_2{R\left(\xi \right)}^2}{C_0+C_{1}R\left(\xi \right)}},$$

(21)

where $a_0,a_1,a_2,$  $C_0$, and $C_1$ are undefined constants to be investigated.

Now, by inserting (21) into Equation (20) and placing the coefficient of the identical power of ${\left(R\left(\xi \right)\right)}^i$, where $(i=0,1,2)$ equal to zero, we can obtain some algebraic equations. Solving these equations, we obtain the value of $a_0,a_1,a_2,C_0,C_1$, and $w$ as follows:

Set 1: $a_0=0$, $a_1=-a_2$, $C_0={\displaystyle \frac{2a_2\left(-10b_2+3\mathrm{i}\sqrt{5}\sqrt{a}k\left(\ln l\right)\sqrt{b_4}\right)}{15a{k}^2{\left(\ln l\right)}^2}}$, $C_1=-{\displaystyle \frac{4\mathrm{i}{a}_2\sqrt{b_4}}{\sqrt{5}\sqrt{a}k\left(\ln l\right)}}$,

$$w={\displaystyle \frac{1}{16}}a{k}^2{\left(-16+(\ln l\right)}^2).$$

(22)

Plugging in (22) along with Equation (10) into (21), we obtain the solution as follows:

$$V_{11}\left(x,t\right)=5^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ik(16x-{\displaystyle \frac{akF}{\alpha }}(-16+{\left(\ln l\right)}^2))}{({\displaystyle \frac{\sqrt{a}k{l}^{k(\frac{x}{2}+\frac{a}{\alpha }F)}\ln l\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(\frac{K}{2\alpha }(x\alpha +2a\mathrm{F})\ln l)a_2}{10\sqrt{a}k(1+l^{k(x+\frac{2aF}{\alpha })})\ln l{b}_0-4i\sqrt{5}{a}_2\sqrt{b_4}}})}^{1/4},$$

(23)

and

$$V_{12}\left(x,t\right)=5^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ik(16x-{\displaystyle \frac{akM}{\alpha }}(-16+{\left(\ln l\right)}^2))}{({\displaystyle \frac{\sqrt{a}k{l}^{k(\frac{x}{2}+\frac{a}{\alpha }F)}\ln l\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(\frac{K}{2\alpha }(x\alpha +2a\mathrm{F})\ln l)a_2}{10\sqrt{a}k(1+l^{k(x+\frac{2aF}{\alpha })})\ln l{b}_0-4i\sqrt{5}{a}_2\sqrt{b_4}}})}^{1/4},$$

(24)

wherein $F={(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\alpha }})}^{\alpha }$.

Set 2: $a_0=0$, $a_1=-a_2$, $C_0=-{\displaystyle \frac{2a_2\left(10b_2+3\mathrm{i}\sqrt{5}\sqrt{a}k\left(lnl\right)\sqrt{b_4}\right)}{15a{k}^2{\left(\ln l\right)}^2}}$, $C_1={\displaystyle \frac{4\mathrm{i}{a}_2\sqrt{b_4}}{\sqrt{5}\sqrt{a}k\left(\ln l\right)}}$,

$$w={\displaystyle \frac{1}{16}}a{k}^2\left(-16+{\left(\ln l\right)}^2\right).$$

(25)

We substitute the values presented in (25) along with Equation (10) into (21). This gives us the following solution:

$$V_{21}\left(x,t\right)={{\displaystyle \frac{15}{\sqrt{2}}}}^{1/4}{\mathrm{e}}^{-{\displaystyle \frac{1}{16}}\mathrm{i}kE}{(-{\displaystyle \frac{\mathrm{i}a{k}^2{l}^{k\left(\frac{x}{2}+\frac{a}{\alpha }\mathrm{F}\right)}{\left(\ln l\right)}^2\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(\frac{k}{2\alpha }(x\alpha +2a\mathrm{F}(\left(\ln l\right))}{-10\mathrm{i}(1+l^{k(x+\frac{2a}{\alpha }\mathrm{F})})b_2+3\sqrt{5}\sqrt{a}\sqrt{b_4}k\left(\ln l\right)(-1+l^{k(x+\frac{2a}{\alpha }\mathrm{F})})}})}^{1/4},$$

(26)

and

$$V_{22}\left(x,t\right)={{\displaystyle \frac{15}{\sqrt{2}}}}^{1/4}{\mathrm{e}}^{-{\displaystyle \frac{1}{16}}\mathrm{i}kE}{(-{\displaystyle \frac{\mathrm{i}a{k}^2{l}^{k\left(\frac{x}{2}+\frac{a}{\alpha }F\right)}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{h}(\frac{k}{2\alpha }\left(x\alpha +2aF\right)\ln l){(\ln l)}^2}{-10\mathrm{i}(1+l^{k(x+\frac{2aF}{\alpha })})b_2+3\sqrt{5}\sqrt{a}\sqrt{b_4}k(-1+l^{k(x+\frac{2a}{\alpha }F)})\ln l}})}^{1/4},$$

(27)

wherein $E=(16x-{\displaystyle \frac{akF}{\alpha }}(-16+{(\ln l)}^2))$, $F={(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\alpha }})}^{\alpha }$.

The solutions presented above contain arbitrary constants. By choosing specific values for these constants, solitons with diverse geometric shapes can be obtained. However, due to page constraints, these solutions are not presented here.

3.2. The Time-Fractional Davey–Stewartson (DS) Equation

In this section, we formulate various soliton solutions to the time-fractional (2 + 1)-dimensional Davey–Stewartson equation using the extended generalized Kudryashov expansion method. We consider the model as shown in [39]:

$$i{V}_t^{\alpha }+{\displaystyle \frac{1}{2}}{\sigma }^2(V_{xx}+{\sigma }^2{V}_{yy})+\lambda {\left|V\right|}^{2}V-V{W}_x=0,$$

$$W_{xx}-{\sigma }^2{W}_{yy}-2\lambda {\left({\left|V\right|}^2\right)}_x=0,$$

(28)

where $V(x,y,t)$ represents the complex wave function (or amplitude) of the surface wave, while $W(x,y,t)$ denotes the real potential function related to this wave. The values of the constants $\sigma$ and $\lambda$ depend on the physical state of the system. The signs and values of $\sigma$ and $\lambda$ are determined by the inherent dispersion and nonlinearity of the system.

To split Equation (28) into its real and imaginary parts, we use the following transformation:

$$V\left(x,y,t\right)=V\left(\xi \right)e^{i\eta },W\left(x,y,t\right)=W\left(\xi \right),$$

(29)

wherein

$$\xi (x,y,t)=x-2\alpha y+{\displaystyle \frac{\alpha }{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta },\eta \left(x,y,t\right)=\alpha x+y+{\displaystyle \frac{k}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }+n,$$

(30)

where $\alpha$ and $k$ are real constants. To split Equation (28) into real and imaginary parts, we substitute (29) and (30) into (28). As a result, we obtain

$${\sigma }^2\left(1+4{\sigma }^2{\alpha }^2\right)V^{″}\left(\xi \right)+2\lambda V^3\left(\xi \right)-\left(2k+{\sigma }^2{\alpha }^2+{\sigma }^4\right)V\left(\xi \right)-2V\left(\xi \right)W^{\prime }\left(\xi \right)=0,$$

(31)

$$\left(1-4{\sigma }^2{\alpha }^2\right)W^{″}\left(\xi \right)=2\lambda \left[V^2\left(\xi \right)\right]\prime .$$

(32)

Integrating (32) with respect to $\xi$, we obtain

$$W^{\prime }\left(\xi \right)={\displaystyle \frac{2\lambda }{\left(1-4{\sigma }^2{\alpha }^2\right)}}{V}^2\left(\xi \right)+C.$$

(33)

Here, $C$ is an integral constant and $\alpha \ne \pm {\displaystyle \frac{1}{2\sigma }}$. Substituting (33) into (31), we obtain

$${\sigma }^2\left(1+4{\sigma }^2{\alpha }^2\right)V^{″}\left(\xi \right)+2\lambda (1-{\displaystyle \frac{2}{\left(1-4{\sigma }^2{\alpha }^2\right)}})V^3\left(\xi \right)-\left(2k+{\sigma }^2{\alpha }^2+{\sigma }^4+2C\right)V\left(\xi \right)=0.$$

(34)

In line with the extended generalized Kudryashov approach, the solution of Equation (34) is

$$V\left(\xi \right)={\displaystyle \frac{a_0+a_{1}R\left(\xi \right)+a_2{R\left(\xi \right)}^2}{b_0+b_{1}R\left(\xi \right)}},$$

(35)

where $a_0$, $a_1$, $a_2$, $b_0$, and $b_1$ are undefined constants to be examined.

Putting (35) into Equation (34) and placing the coefficient of the alike power of $R^i\left(\xi \right)$, where $\left(i=0,1,2\right)$ equal to zero, we obtain some algebraic equations. By solving these equations, we obtain the value of $a_0$, $a_1$, $a_2$, $b_0$, $b_1$, and $C$.

Set 1: $a_0=0$, $a_1={\displaystyle \frac{\mathrm{i}\sigma lnl{b}_1\sqrt{-1+4{\alpha }^2{\sigma }^2}}{2\sqrt{\lambda }}}$, $a_2=-{\displaystyle \frac{\mathrm{i}\sigma lnl{b}_1\sqrt{-1+4{\alpha }^2{\sigma }^2}}{\sqrt{\lambda }}}$, $b_0=0$,

$$C={\displaystyle \frac{1}{4}}(-2(2k+{\alpha }^2{\sigma }^2+{\sigma }^4)-({\sigma }^2+4{\alpha }^2{\sigma }^4\left){\left(lnl\right)}^2\right).$$

(36)

By substituting the values from Equation (36) into Solution (35) along with Equation (30), we obtain the following solution:

$$V_{31}\left(x,y,t\right)={\displaystyle \frac{{\mathrm{ie}}^{\mathrm{i}(\alpha x+y+\frac{k}{\beta }{\left(t+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\right)}^{\beta }+n)}lnl\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}}{2\sqrt{\lambda }}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left({\displaystyle \frac{\xi }{2}}lnl\right).$$

(37)

and

$$V_{32}\left(x,y,t\right)={\displaystyle \frac{{\mathrm{ie}}^{\mathrm{i}(\alpha x+y+\frac{k}{\beta }{\left(t+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\right)}^{\beta }+n)}lnl\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}}{2\sqrt{\lambda }}}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left({\displaystyle \frac{\xi }{2}}lnl\right).$$

(38)

Set 2: $a_0=-{\displaystyle \frac{1}{4\lambda b_0{b}_1+2\lambda b_1^2}}\left(2\mathrm{i}\sqrt{\lambda }\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}lnl{b}_0^3+\mathrm{i}\sqrt{\lambda }\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}lnl{b}_0^2{b}_1+\sqrt{-\lambda {\sigma }^2\left(-1+4{\alpha }^2{\sigma }^2\right){\left(lnl\right)}^2{b}_0^2{\left(2b_0^2+3b_0{b}_1+b_1^2\right)}^2}\right)$, $a_1=-{\displaystyle \frac{\mathrm{i}\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}lnl\left(2b_0+b_1\right)}{2\sqrt{\lambda }}}$, $a_2=0$,

$$C={\displaystyle \frac{1}{4}}(-2(2k+{\alpha }^2{\sigma }^2+{\sigma }^4)-({\sigma }^2+4{\alpha }^2{\sigma }^4\left){\left(lnl\right)}^2\right).$$

(39)

By using Equation (39) together with Equation (35) in Equation (29), we derive the following solution:

$$V_{41}\left(x,y,t\right)={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\varphi [x,y,t]}}{b_0+\frac{b_1}{1+l^{\xi }}}}({\displaystyle \frac{\mathrm{i}\sigma \sqrt{g}lnl(2b_0+b_1)}{2(1+l^{\xi })\sqrt{\lambda }}}+{\displaystyle \frac{2\mathrm{i}\sqrt{\lambda }\sigma \sqrt{g}lnl{b}_0^3+\mathrm{i}\sqrt{\lambda }\sigma \sqrt{g}lnl{b}_0^2{b}_1+h}{4\lambda b_0{b}_1+2\lambda b_1^2}}).$$

(40)

and

$$V_{42}\left(x,y,t\right)={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\varphi [x,y,t]}}{b_0+\frac{b_1}{1-l^{\xi }}}}({\displaystyle \frac{\sigma \sqrt{g}lnl(2b_0+b_1)}{2(-1+l^{\xi })\sqrt{\lambda }}}-{\displaystyle \frac{2\sqrt{\lambda }\sigma \sqrt{g}lnl{b}_0^3+\mathrm{i}\sqrt{\lambda }\sigma \sqrt{g}lnl{b}_0^2{b}_1+h}{4\lambda b_0{b}_1+2\lambda b_1^2}}),$$

(41)

where $g=1-4{\alpha }^2{\sigma }^2$ and $h=\sqrt{-\lambda {\sigma }^{2}g{\left(lnl\right)}^2{b}_0^2{(2b_0^2+3b_0{b}_1+b_1^2)}^2}$.

The solutions established above include subjective constants. By selecting particular values for these constants, it is possible to obtain solitons with various geometric shapes. However, these solutions are not shown here due to space limitations.

4. Graphical Representations and Physical Explanations

In this section, we discuss the graphical representation and interpretation of the deterministic solutions obtained from the time-fractional cubic-quintic-septic-nonic equation and the time-fractional DS equation. We depict the 3D and 2D profiles of these solutions by selecting specific values for various parameters. This helps us understand how the mechanisms of the physical phenomena work.

4.1. The Nonlinear Time-Fractional Cubic-Quantic-Septic-Nonic Equation

In this part, we provide a physical explanation and demonstration of the solution to the nonlinear cubic-quintic-septic-nonic equation. We use graphs to graphically represent the solutions. Various details in the mathematical sciences are also presented, mainly for the purpose of comparison. The 3D graphics explore the behavior of the wave in terms of spatial and temporal coordinates $\left(x,t\right),$ and the two-dimensional graphs are plotted for different values of fractional order to describe the effect of fractional-order derivatives on solitary waves. These graphs are illustrated by using Wolfram Mathematica. In Figure 1, the modulus plot of $V_{11}(x,t)$ is depicted with the values of the parameters $k=0.9$, $w=3$, $l=2$, $a=2$, $a_2=-0.0006$, $C_0=-0.0001$, $C_2=-0.0003$, and $b_4=-0.00015$ within the intervals $-20\le x\le 20$ and $0\le t\le 10$. When $\alpha =0.1$, the wave is initially started in the form of an asymptotic shape. For the change in the value of $\alpha$ (fractional-order) the soliton wave becomes a bell-shape soliton, i.e., the soliton transforms the shape from its initial state for $\alpha =0.1$ to a bell shape for $\alpha =0.9.$ We observe that the soliton is moved from the left direction to the right direction. Also, the shape of this soliton is rapidly changed when $\alpha >0.1$.

In Figure 2, the modulus plot of the solution $V_{11}(x,t)$ is depicted with the values of parameters $k=0.03, w=3, a=1,b_2=-0.03, b_4=-0.0015, l=6, C_0=0.0001,$ and $a_2=-0.00002$ within the intervals $-20\le x\le 20$ and $0\le t\le 10$. Initially, the tail of the soliton cut the $x$-axis at (40, 0) approximately, and the soliton cut the $y$-axis at (0, 0.01) for the value of fractional-order $\alpha =0.03$. When $\alpha =0.1$, the soliton cut the $y$-axis at the point (0, 0.35). We observe from $2\left(a\right)$ that for the change in the value of fractional order, the shape of the soliton is gradually changed.

The plot of the modulus of the solution $V_{11}(x,t)$ reveals a deformed bell-shape soliton for $k=2,w=1,l=4,a=2,a_2=-10,C_0=-0.000009, b_2=0.007$, and $b_4=-10$ within the intervals $-10<x<20$ and $0<t<10$, which is presented in Figure 3. From the figure, it is observed that initially, the shape of the soliton is the asymptotic type at $\alpha =0.1$. As the value of the fractional order $\left(\alpha \right)$ gradually increases, the graph slowly changes into a deformed bell-shaped soliton. For $\alpha =0.2$, we observe an approximately stable solitary wave. After that, the wave travels from the left to right direction at a constant speed, maintaining its shape and size.

The modulus plot of V₁₂(x,t) exhibits a soliton, which is portrayed in Figure 4 by selecting the parameters $k=2$, $w=1$, $l=2$, $a=0.000012$, $a_2=0.00002$, $C_0=0.00003$, $b_2=0.003$, and $b_4=0.000015$ within the intervals $-10\le x\le 20$ and $0\le t\le 100$. When $\alpha =0.1$, the wave starts with an asymptotic shape. As the value of $\alpha$ (fractional-order) changes, the soliton transforms into a deformed spike shape. Specifically, it changes from an asymptotic shape at $\alpha =0.1$ to a spike shape at $\alpha =0.9$. We notice that the soliton moves from left to right, and its shape quickly changes when $\alpha$ is greater than 0.1.

The modulus plot of V₂₁(x,t) exhibits a cuspon, represented in Figure 5 by selecting the parameters $k=1$, $w=1$, $l=4$, $a=0.1$, $a_2=0.3$, $b_2=-0.003$, $C_0=0.1$, and $b_4=0.01$ inside the range of $-10\le x\le 20$ and $0\le t\le 10$. For $\alpha =0.01$, the wave initially begins in an asymptotic shape. While the value of $\alpha$ (fractional-order) changes, the soliton changes into a cuspon-type soliton, i.e., the soliton transforms from an asymptotic shape for $\alpha =0.01$ to a cuspon shape for $\alpha =0.9$. We observe that the soliton is moved from the left direction to the right direction, and the shape of this soliton is rapidly changed when $\alpha >0.01$.

The modulus plot of solution V₂₁(x,t) exhibits a parabolic shape soliton, as showed in Figure 6 by selecting the parameters $k=0.03$, $a=0.02$, $l=4$, $a_2=0.3$, $C_0=0.1$, $b_2=-0.003$, and $b_4=0.01$ within the interval $-20\le x\le 20$ and $0\le t\le 10$. When $\alpha =0.001$, the wave is initially like a plane. As the value of $\alpha$ (fractional-order) changes, the soliton transforms into a parabolic shape. It changes from a plane for $\alpha =0.001$ to a parabolic-shaped soliton for $\alpha =0.3$. We observe that the shape of the soliton changes quickly when $\alpha >0.001$.

The plot of the modulus of solution V₂₂(x,t) exhibits a squeezed bell-type soliton, presented in Figure 7 by choosing the parameters $k=7$, $w=1$, $l=6$, $a=0.001$, $a_2=3$, $C_0=3$, $b_2=-0.003$, and $b_4=0.0002$ within the interval $-10\le x\le 20$ and $0\le t\le 10$. When $\alpha =0.0001$, initially, the wave shape is a one-sided squeezed soliton. For the change in the value of $\alpha$ (fractional-order), the soliton wave becomes a squeezed bell-type soliton, i.e., the soliton transforms the shape from its initial state (a one-sided squeezed soliton) for $\alpha =0.0001$ to a squeezed bell-type soliton for $\alpha =0.05.$ We observe that the shape of this soliton is rapidly changed when $\alpha >0.0001$.

4.2. The Nonlinear Time Fractional Davey–Stewartson Equation

The graph of the modulus plot of solution V₃₁(x,t) is exhibited in Figure 8 by selecting the parameters $k=1$, $\alpha =0.1$, $\sigma =1$, $\lambda =1$, $l=4$, $t=1$, $b_0=1$, and $b_1=1$. Here, for the 3D graph, we put a fixed value for $t$, and the intervals of $x$ and $y$ are $-10\le x,y\le 10$. Also, for the 2D graph, space is constant in Figure 8d, and time is constant in Figure 8e, i.e., the graph in Figure 8d is drawn on a plane, and Figure 8e is also drawn on a plane. When $\beta =0.1$, the wave is initially started and after a gradual change in the fractional order $\left(\beta \right)$, the wave shape is completed, slowly corresponding to the value of $\beta$. Here, we observe that the shape of this soliton becomes a lump shape at $\beta =0.9$.

In Figure 9, the modulus plot of solution $V_{32}(x,t)$ is depicted for the values of parameters $k=1$, $\alpha =0.1$, $\sigma =1$, $\lambda =1$, $l=4$, $t=1$, $b_0=1$, and $b_1=1$. Here, for a 3D graph, we consider $t$ as the constant, and the values of $x$ and $y$ are inside the intervals $0\le x\le 10$ and $-10\le y\le 10$. Also, for the 2D graph, time is constant in Figure 9d, and one space dimension is also considered constant in Figure 9e. Initially, for $\beta =0.04$, the soliton starts in an asymptotic shape at $\alpha =0.03$. For the change in the value of $\beta$, the soliton wave becomes an anti-bell-shaped soliton. The soliton transforms the shape rapidly from its initial state (asymptotic) for $\beta =0.04$ to an anti-bell shape for $\beta =0.2$. From the above discussion, we can say that the shape of this soliton rapidly changes for $\beta >0.04$.

In Figure 10, the modulus plot of $V_{33}(x,t)$ is depicted for the values of the parameters $k=1$, $\alpha =1$, $l=2$, $\sigma =-0.01$, $\lambda =0.001$, $b_0=-0.1$, and $b_1=1$. Here, the soliton solution is the singular type for all values of fractional order $\beta$. For various arbitrary values of the parameters, we obtain additional different graphics even though they have similar meanings. Therefore, for brevity, a few solutions are presented.

5. Comparison of the Results

In this section of the article, we compare the results obtained with those already available in the literature. In

Table 1. Comparison of the results of the cubic-quintic-septic-nonic equation.

Solutions Obtained by the Present Study	Solutions Obtained by Samir et al. [1]
$V_{11}\left(x,t\right)=5^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ikE}$ ${\left({\displaystyle \frac{\sqrt{a}k{l}^{k(\frac{x}{2}+\frac{a}{\alpha }F)}\ln l\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{K}{2\alpha }\right(x\alpha +2a\mathrm{F}\left)\ln l\right)a_2}{10\sqrt{a}k(1+l^{k(x+\frac{2aF}{\alpha })})\ln l{b}_0-4i\sqrt{5}{a}_2\sqrt{b_4}}}\right)}^{1/4}$	$q\left(x,t\right)=e^{i\left(-kx+wt+\theta \right)}$ ${\left({\displaystyle \frac{3(w+a{k}^2}{b_2}}(1\pm \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}[2\sqrt{{\displaystyle \frac{w+a{k}^2}{a}}}\left(x-vt\right)]\right)}^{1/4}$.
$V_{12}\left(x,t\right)=5^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ikE}$ ${\left({\displaystyle \frac{\sqrt{a}k{l}^{k(\frac{x}{2}+\frac{a}{\alpha }F)}\ln l\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{K}{2\alpha }\right(x\alpha +2a\mathrm{F}\left)\ln l\right)a_2}{10\sqrt{a}k(1+l^{k(x+\frac{2aF}{\alpha })})\ln l{b}_0-4i\sqrt{5}{a}_2\sqrt{b_4}}}\right)}^{1/4}$	$q\left(x,t\right)=e^{i\left(-kx+wt+\theta \right)}$ ${\left({\displaystyle \frac{3(w+a{k}^2}{b_2}}(1\mp \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}[2\sqrt{{\displaystyle \frac{w+a{k}^2}{a}}}\left(x-vt\right)]\right)}^{1/4}$.
$V_{21}\left(x,t\right)={{\displaystyle \frac{15}{\sqrt{2}}}}^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ikE}$ ${\left({\displaystyle \frac{-ia{k}^2{l}^{k\left(\frac{x}{2}+\frac{a}{\alpha }F\right)}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{k}{2\alpha }\left(x\alpha +2aF\right)\ln l\right){\left(\ln l\right)}^2}{-10i(1+l^{k(x+\frac{2aF}{\alpha })})b_2+3\sqrt{5}\sqrt{a}\sqrt{b_4}k(-1+l^{k(x+\frac{2a}{\alpha }F)})\ln l}}\right)}^{1/4}$	$q\left(x,t\right)=e^{i\left(-kx+wt+\theta \right)}$ ${\left({\displaystyle \frac{3(w+a{k}^2}{b_2}}(1\pm \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}[2\sqrt{{\displaystyle \frac{w+a{k}^2}{a}}}\left(x-vt\right)]\right)}^{1/4}$.
$V_{22}\left(x,t\right)={{\displaystyle \frac{15}{\sqrt{2}}}}^{1/4}{e}^{-{\displaystyle \frac{1}{16}}ikE}$ ${\left({\displaystyle \frac{-ia{k}^2{l}^{k\left(\frac{x}{2}+\frac{a}{\alpha }F\right)}Cosch\left(\frac{k}{2\alpha }\left(x\alpha +2aF\right)\ln l\right){\left(\ln l\right)}^2}{-10i(1+l^{k(x+\frac{2aF}{\alpha })})b_2+3\sqrt{5}\sqrt{a}\sqrt{b_4}k(-1+l^{k(x+\frac{2a}{\alpha }F)})\ln l}}\right)}^{1/4}$	$q\left(x,t\right)=e^{i\left(-kx+wt+\theta \right)}$ ${\left({\displaystyle \frac{3(w+a{k}^2}{b_2}}(1\mp \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}[2\sqrt{{\displaystyle \frac{w+a{k}^2}{a}}}\left(x-vt\right)]\right)}^{1/4}$.

, we compare the results of the cubic-quintic-septic-nonic equation with the results of Samir et al. [1]. This comparison helps us verify the accuracy and reliability of this study. On the other hand, in

Table 2. Comparison of the results of the Davey–Stewartson (DS) equation.

Solutions Obtained by the Present Study	Solutions Obtained by Mirzazadeh [39]
$V_{31}\left(x,y,t\right)={\displaystyle \frac{i{e}^{i(\alpha x+y+\frac{k}{\beta }{\left(t+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\right)}^{\beta }+n)}\ln l\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}}{2\sqrt{\lambda }}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left({\displaystyle \frac{\xi }{2}}\ln l\right)$	$U\left(x,y,t\right)=\pm e^{i\eta \left(x,y,t\right)}$ $\left(2\sigma \ln a\sqrt{{\displaystyle \frac{-1+4{\alpha }^2{\sigma }^2}{\lambda }}}\mathrm{csch}\left(\xi \ln l\right)\right)$
$V_{32}\left(x,y,t\right)={\displaystyle \frac{i{e}^{i(\alpha x+y+\frac{k}{\beta }{\left(t+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\right)}^{\beta }+n)}\ln l\sigma \sqrt{-1+4{\alpha }^2{\sigma }^2}}{2\sqrt{\lambda }}}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left({\displaystyle \frac{\xi }{2}}\ln l\right)$	$U\left(x,y,t\right)=-e^{i\eta \left(x,y,t\right)}$ $\left({\displaystyle \frac{\sigma \ln a}{2}}\sqrt{{\displaystyle \frac{-1+4{\alpha }^2{\sigma }^2}{\lambda }}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\right({\displaystyle \frac{\xi }{2}}\ln l\left)\right)$
$V_{41}\left(x,y,t\right)={\displaystyle \frac{e^{i\varphi \left[x,y,t\right]}}{b_0+\frac{b_1}{1+l^{\xi }}}}({\displaystyle \frac{i\sigma \sqrt{g}\ln l\left(2b_0+b_1\right)}{2\left(1+l^{\xi }\right)\sqrt{\lambda }}}+{\displaystyle \frac{2i\sqrt{\lambda }\sigma \sqrt{g}\ln l{b}_0^3+i\sqrt{\lambda }\sigma \sqrt{g}\ln l{b}_0^2{b}_1+h}{4\lambda b_0{b}_1+2\lambda b_1^2}})$	$U\left(x,y,t\right)=-e^{i\eta \left(x,y,t\right)}$ $\left({\displaystyle \frac{\sigma \ln a}{2}}\sqrt{{\displaystyle \frac{-1+4{\alpha }^2{\sigma }^2}{\lambda }}}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\right({\displaystyle \frac{\xi }{2}}\ln l\left)\right)$
$V_{42}\left(x,y,t\right)={\displaystyle \frac{e^{i\varphi \left[x,y,t\right]}}{b_0+\frac{b_1}{1-l^{\xi }}}}({\displaystyle \frac{\sigma \sqrt{g}\ln l\left(2b_0+b_1\right)}{2\left(-1+l^{\xi }\right)\sqrt{\lambda }}}-{\displaystyle \frac{2\sqrt{\lambda }\sigma \sqrt{g}\ln l{b}_0^3+i\sqrt{\lambda }\sigma \sqrt{g}\ln l{b}_0^2{b}_1+h}{4\lambda b_0{b}_1+2\lambda b_1^2}})$	No such solution found.

, we compare the results obtained for the Davey–Stewartson equation with the results obtained by Mirzazadeh [39]. The comparison demonstrates the accuracy of the obtained results and the effectiveness of the method in solving complex equations.

From the analysis of the information in Table 1 and Table 2, it is clear that this study has established several new solutions that were not found in previous research. Furthermore, this study focuses on fractional-order models, while earlier studies were based on integer-order models. As a result, the findings of this research could be useful to analyze data transmission systems.

6. Conclusions

In this study, we investigated two nonlinear time-fractional models, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, using the extended generalized Kudryashov technique, and found some fresh and typical solutions. The fractional derivative is considered in the sense of a beta derivative since it follows all the characteristics of the classical derivative. The solutions obtained revealed a range of rich structured solitons, including bell-shaped, parabolic, cuspon, anti-bell-shaped, and singular solitons. The 3D and 2D graphs show that the fractional-order derivative significantly impacts soliton dynamics and offers new insights into the behavior of nonlinear models. The beta fractional derivative is important because it offers a balance between simplicity and the ability to generalize classical integer-order derivative, has an impact on fractional dynamics on soliton solutions in nonlinear models, and provides new insights into maintaining the mathematically manageable structure. This study demonstrates the effectiveness of fractional derivatives in expanding the understanding of complex nonlinear systems, which could be applied in future studies to other fractional models in fields such as optical fiber communication and plasma physics.