Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity: Model problems

1.   Introduction

This study begins by setting the targeted model in the context of its earlier research and thoroughly analyzing pertinent literature. This part provides an overview of the study's main goals, identifies gaps in the existing literature, and includes information on the manuscript's organizational structure.

Nonlinear partial differential equations (NPDEs) have become increasingly prominent as fundamental study fields in recent decades. This is particularly true for discovering new characteristics of intricate phenomena in various scientific domains, such as atomic, particle, optical, nuclear, and biological physics ^[1,2,3]. To demonstrate various engineering, physics, and natural processes, several nonlinear models have been developed ^[4,5]. Several methods and strategies have been developed to investigate numerical, analytical, and semi-analytical solutions. Spline methods ^[6], finite difference method ^[7], the Adomian decomposition method ^[8], the variational iteration method ^[9], finite eliminating method ^[10], the ($\frac{G'}{G}$)-expansion method ^[11], the Riccati expansion method ^[12], the sine-cosine expansion method ^[13], the tanh-expansion method ^[14], the modified simple equation method ^[15], and the sech-tanh expansion method ^[16]. Over the last couple of years, analysis of nonlinear dynamic systems and soliton has attracted much interest because of its broad applications in areas as diverse as optics and fluid mechanics and material sciences ^[17,18]. Analytical solutions of nonlinear differential equations including generalized systems, algebraic constraints, and different fractional, and perturbation models has often been found to capture the physical behavior as the wave, the fluid, and the electromagnetic vibrations ^[19,20]. To obtain more information, one used exact solutions and methods such as Bäcklund transformation, bifurcation analysis, and observer design in these systems. This paper extends the current literature by emphasizing the improvement of solutions? accuracy and versatility in different mathematical and natural science problems through the use of sophisticated mathematical tools, namely fuzzy logic and fractional calculus ^[21,22,23]. We hope that by applying these techniques in conjunction with existing approaches, enhanced models are created that enable better analysis and decision-making in situations characterized by risk and ambiguity ^[24,25].

It is important to investigate explicit solutions for NPDEs using analytical techniques, and it is crucial to comprehend the behaviors of several elements in various scientific fields ^[26,27,28]. Soliton solutions, among the explicit solutions to NPDEs, remain significant from an academic perspective because they offer more breadth and depth than conventional solutions. A soliton is a single, autonomous wave packet that moves across a medium without altering its shape or velocity ^[29,30,31]. They are valuable in many technological and scientific fields because they have inherent stability and longevity. They provide effective information transport and far-reaching coherence preservation for nonlinear systems. To obtain novel soliton solution outcomes, mathematicians have created powerful methods. Among these noteworthy techniques are the Khater methods ^[32], Sin-Gordon method ^[33], Poincaré-Lighthill-Kuo method ^[34], exp-function method ^[35], ($\frac{G'}{G}$)-expansion method ^[36,37], Kudryashov method ^[38], Riccati-Bernoulli Sub-ODE ^[39], Sardar sub-equation method ^[40], sub-equation method ^[41], extended direct algebraic method (EDAM) ^{[42,43,44,45]}, and Hirota's bilinear method ^[46]. These incredibly efficient techniques, aimed at creating soliton solutions for NPDEs, are constantly evolving. They are a critical development designed to produce new analytical solutions for NPDEs. Traveling wave patterns inside NPDEs are of great interest in many fields, such as physics, fluid mechanics, and engineering and other scientific fields ^{[47,48,49,50,51]}.

In modern mathematics and physics, gauge equivalence is used for relating some integrable nonlinear evolution equations (NEEs) to each other or, in fact, in a quite different manner. It is known that the nonlinear Schrödinger equation is the gauge partner of the Heisenberg ferromagnetic equation. Gauge equivalencies allow us to exploit the relationship between integrable nonlinear equations. Using these equivalencies, we can obtain crucial insights into each equation from its gauge partner. This is achieved by transforming solutions from one equation into solutions from another ^[52,53,54]. In a formal approach, the system may be represented by two equivalent zero curvature criteria, corresponding to two different sets of Lax operators ^[55,56]. This connection has been thoroughly examined in the previously cited literature and other pertinent sources. This characteristic of the gauge transformation has been investigated to build the gauge equivalent counterpart of the Zhaidary equation and to derive integrable reductions and generalizations. The Zhaidary equations allow for several integrable reductions, including the integrable Kuralay-Ⅱ system (K-IIS), Shynaray-Ⅱ and Zhanbota-Ⅱ. Among these, the K-IIS, found in ^[57,58,59], has been considered in this research. This model is articulated as:

where $i = \sqrt{-1}$, $q = q(x, t)$ as a real valued function and $p = p(x, t)$ as a complex valued function, $x$ and $t$ are real spatiotemporal variables, and $\mu = \pm 1$. This model has applications in many different disciplines of optics and physics.

Numerous other researchers have addressed K-IIS in integer and fractional form with different mathematical tools before this research survey. For instance, by employing the Hirota bilinear approach, Sagidullayeva et al. ^[57] investigated this model in 2022 and determined a gauge equivalency between them. Faridi et al. ^[58] employed the innovative approach of auxiliary equations to analyze the integrable motion of induced space curves and deduce various soliton solutions. To look at exact solutions of the K-IIS, such as solitary waves and optical solitons, Mathanaranjan ^[59] used the new extended auxiliary equation and modified F-expansion approaches. Furthermore, the modulation instability gain spectrum was obtained by applying linear stability analysis to modulation instability analysis. Exact solutions of the fractional K-IIS were investigated by Zafar et al. ^[60] in 2023 using a variety of techniques, including the extended sinh-Gordon equation expansion, the generalized Kudryashov, and exp-function schemes. Using the Jacobi elliptic function expansion approach, Khan et al. ^[61] looked into the precise solitary wave solutions to the truncated fractional K-IISs. These investigations' soliton solutions have potential uses in the scientific and technical domains.

Previous studies on the exploration of the solitonic phenomena in K-IIS by numerous investigators make it evident that optical fractal solutions have not yet been investigated. This claim highlights a substantial gap in the existing corpus of research. Our paper seeks to address this gap by offering a thorough analysis of the model and outlining the suggested technique: the Riccati modified extended simple equation method (RMESEM).

This study's goals and objectives are as follows. The intended K-IIS will first be transformed into a single, more manageable nonlinear ordinary differential equation (NODE) via a wave transformation. Next, we will convert the NODE into an algebraic system of equations by assuming a closed form solution using the RMESEM technique. Ultimately, the system will be examined using the Maple tool to determine the K-IIS optical soliton solutions. In conclusion, we will analyze how temporal variation and free parameters affect the model's optical soliton solutions, illustrating a few of these solutions with a combination of contour, two-, and three-dimensional visualizations. Additionally, we will demonstrate how the instability caused by periodic-axial perturbation in complex solutions results in derived solitons solutions that resemble fractals, while real solutions, as defined by K-IIS, resemble hump solitons.

The remaining sections are organized as follows. The analytical processes of the RMESEM are explained in Section 2. K-IIS is addressed and discussed in Section 3 to generate novel optical soliton solutions. A graphic depiction of the propagating behavior of the generated optical solitons is given in Section 4. The results are outlined in Section 5, and the final section includes the appendix.

2.   The functional strategy of Riccati modified extended simple equation method (RMESEM)

In order to study soliton phenomena in nonlinear models, numerous analytical approaches have been established in the literature. The functioning process of the upgraded RMESEM is described in this section. Examining the resulting generic NPDE ^[62]:

where $\mathfrak{p} = \mathfrak{p}(t, y_1, y_2, y_3, \ldots, y_r)$.

The procedures listed below will be followed in order to solve (2.1):

(1) The variable-form complex transformation $\mathfrak{p}(t, y_1, y_2, y_3, \dots, y_r) = \mathfrak{P}({\varrho})$ is first performed. For ${\varrho}$, several representations are known. By useing this technique, Eq (2.1) is transformed to get the subsequent NODE:

where $\mathfrak{P}' = \frac{d \mathfrak{P}}{d{\varrho}}$. The homogeneous balance condition (2.2) may be on occasion enforced on the NODE with the use of the integrating equation.

(2) Next, utilizing the solution provided by the extended Riccati-NODE, the resulting finite series-based solution for the NODE in (2.2) is recommended:

Here, the solution to the resultant extended Riccati-NODE is denoted by ${B}({\varrho})$, and the unidentified constants required to be found later are represented by the variables ${F}_{j} ({j} = 0, ..., {s})$ and ${S}_{{j}} ({{j}} = 0, ..., {s}-1)$.

where ${\lambda}, {\eta}$ and ${\nu}$ are constants.

(3) We may get the positive integer ${s}$ needed in Eq (2.3) by homogeneously balancing the greatest nonlinear component and the highest-order derivative in Eq (2.2).

(4) All the components of ${B}({\varrho})$ are then combined into an equal ordering when (2.3) is inserted into (2.2) or the equation that emerges from the integration of (2.2). When this process is used, an equation in terms of ${B}({\varrho})$ is generated. By setting the coefficients in the resulting equation to zero, one may get an algebraic system of equations representing the variables ${F}_{j} ({j} = 0, ..., {s})$ and ${S}_{{j}} ({{j}} = 0, ..., {s}-1)$ along with additional accompanying parameters.

(5) With Maple, the set of nonlinear algebraic equations is analytically evaluated.

(6) To acquire analytical soliton solutions for (2.1), the next step is to compute and enter the unidentified values in addition to ${B}({\varrho})$ (the Eq (2.4) solution) in Eq (2.3). By using (2.4)'s general solution, we might potentially derive a multitude of soliton solutions.

Following is an illustration of these clusters.

3.   Execution of RMESEM and construction of optical soliton solutions

In this study, we present optical soliton solutions for Eq (2.1) by utilizing the suggested RMESEM. We start with the complex wave transformation that follows:

After replacing (3.1) in (2.1) and partitioning the imaginary and real parts, we get:

One time integration w.r.t $\varrho$ of the second part in (3.2) with zero constant of integration yields:

Putting (3.3) in (3.2) reduces the real part of the entire system to the ensuing single NODE:

with the constraint condition from the imaginary part:

Using (3.4) to determine the homogeneous balancing principle within $-\varsigma \delta P''$ and $-\frac{2 \mu \delta P^3}{\varsigma}$, ${s} = 1$ is recommended. Inputting ${s} = 1$ into (2.3) yields the following series solution for Eq (3.4):

By inserting (3.6) into (3.4) and collecting every single term with the same ordering of $B(\varrho)$, an expression in $B(\varrho)$ is obtained. By setting the coefficients to zero, the statement is simplified to a system of nonlinear algebraic equations. Using Maple to solve the resulting problem, the following three types of solutions are found:

Case 1.

Case 2.

Assuming Case 1, we acquire the subsequent clumps of optical soliton solutions for K-IIS stated in (1.1):

Clump 1.1. In the case of ${N} < 0, \; {\nu}\neq0$,

and

Clump 1.2. In the case of ${N} > 0, \; {\nu}\neq0$,

and

Clump 1.3. In the case of ${N} = 0, \; {\eta}\neq0$,

Clump 1.4. In the case of ${N} = 0$, in case when ${\eta} = {\nu} = 0$,

Clump 1.5. In the case of ${\eta} = \varpi$, ${\lambda} = h\varpi(h\neq0)$ and ${\nu} = 0$,

In the above optical soliton solutions of Case 1,

Assuming Case 2, we acquire the subsequent clumps of optical soliton solutions for K-IIS stated in (1.1):

Clump 2.1. In the case of ${N} < 0, \; {\nu}\neq0$,

and

Clump 2.2. In the case of ${N} > 0, \; {\nu}\neq0$,

and

Clump 2.3. In the case of ${N} = 0, \; {\eta}\neq0$,

Clump 2.4. In the case of ${N} = 0$, in case when ${\eta} = {\lambda} = 0$,

Clump 2.5. In the case of ${\eta} = \varpi$, ${\lambda} = h\varpi(h\neq0)$ and ${\nu} = 0$,

Clump 2.6. In the case of ${\eta} = \varpi$, ${\nu} = h\varpi(h\neq0)$ and ${\lambda} = 0$,

Clump 2.7. In the case of ${\lambda} = 0$, ${\nu}\neq0$ and ${\eta}\neq0$,

and

In the above optical soliton solutions of Case 2,

4.   Discussion

In the present section of the study, we provide the frameworks of the several optical wave types that are included in the model. Using RMESEM, we were able to extract and visually represent the wave patterns of optical solitons in 3D, contour, and 2D forms. Comprehending the behavior of related physical events requires these understandings. The discovered optical soliton solutions are predicted to significantly expand our understanding of the dynamics of optical pulse theory in optical fibers. There might be several beneficial uses for these produced optical solitons in the telecom industry. Moreover, our suggested RMESEM illustrates its use by expanding the range of optical soliton solutions, providing significant understanding into the K-IIS dynamics, and indicating potential uses in nonlinear model management.

4.1.   The dynamics of p(x, t)

This subsection presents images of a few artificially created optical solitons within the framework of K-IIS complex solutions, (Figures 1–6). The purpose of the images is to provide a visual representation of how these objects lose stability as they get closer to an axis, displaying periodic-axial disturbances and eventually forming optical fractals. In the context of K-IIS, an optical soliton is a self-sustaining pattern that moves across fiber optic medium without altering its form or speed. When these waves interact or collide with other waves of a similar kind, they are well known for their ability for rebuilding and maintaining themselves. However, our investigation revealed that the dispersion of nonlinearity and the presence of external input lead to axial and periodic disturbances in the created solitons, which in turn cause instabilities in the soliton and lead to the production of optical fractals. Nevertheless, the soliton interaction (especially with lumps) and self-resemblance have also contributed to the formation of the fractal structure in some of our solitons.

4.2.   The dynamics of q(x, t)

In order to illustrate that the real solutions, in the framework of K-IIS, adopt the shape of hump solitons, several developed optical solitons are presented in this subsection, (Figures 7–12). A localized wave packet that keeps its form and speed while moving through a medium is called a hump optical soliton. This unique variety of kink soliton features peaks one or more during light hump and drops during dark hump. These solitons arise in models with modulation instability and are caused by the action of the nonlinear medium. The hump may develop in the presence of disturbances because these wave structures are stable and localized. Hump optical solitons are a vital field of study with many applications in fiber and nonlinear optics, where regulating light propagation plays a key role. The efficiency of optical systems can be improved by the manipulation and generation of such hump optical soliton.

5.   Conclusions

In this study, we have generated some precise optical soliton solutions for K-IIS by using the effective RMESEM. We were able to find solutions that were trigonometric, rational, hyperbolic, exponential, and hyperbolic. To aid in your understanding of the propagation behaviors of the generated optical solitons, we have provided extra 3D, contour, and 2D charts for the free selections of the physical parameters. The obtained solitons solutions take the form of fractals due to the instability caused by periodic-axial perturbation in complex solutions, whereas the real solutions, within the context of K-IIS, take the form of hump solitons, as these visuals demonstrate the graphic behaviors of several optical solitons. In the telecom sector, the generated optical solitons have a number of useful applications. By extending the spectrum of optical soliton solutions, offering insightful information on the K-IIS dynamics, and suggesting possible applications in nonlinear model management, our recruited RMESEM further demonstrates its value. Although the RMESEM has greatly advanced our knowledge of soliton dynamics and how they relate to the models that are being studied, it is crucial to recognize the limits of this approach, especially in situations where the nonlinear component and largest derivative are not equally balanced. Despite this drawback, the current study shows that the technique used in this work is highly productive, trustworthy, and adaptable for nonlinear issues in a range of natural scientific fields. Further research on the soliton's stability, fractal solitons' sensitivity, the inclusion of fractional derivatives and their effects on fractal solitons, and the computation of fractal theory's scaling factors are the future's goals of this project.

Motivated by our findings, we anticipate that further trials will validate the soliton formations we have estimated, including the fascinating fractal structures, determined by our analysis of the K-IIS. We anticipate that similar motifs with hump solitons may manifest in optical studies, in which solitons have already been detected, particularly if axial-periodic perturbations are used. Our contour, 2D, and 3D representations make it easy to recognize these formations, and our hypothesis may be confirmed by comparable practical configurations. This work paves the way for valuable experimental confirmation in optical structures while also improving our theoretical understanding of theory.

Author contributions

Azzh Saad Alshehry: conceptualization, formal analysis, investigation, methodology, project administration, supervision, validation, writing-original draft, writing-review and editing; Safyan Mukhtar: data curation, investigation, software, visualization, writing-original draft; Ali M. Mahnashi: formal analysis, methodology, investigation, software, supervision, validation. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU241810).

Funding

This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (KFU241810).

Conflict of interest

The authors declare that they have no conflicts of interest.