Applications of Riccati–Bernoulli and Bäcklund Methods to the Kuralay-II System in Nonlinear Sciences

Abstract

The Kuralay-II system (K-IIS) plays a pivotal role in modeling sophisticated nonlinear wave processes, particularly in the field of optics. This study introduces novel soliton solutions for the K-IIS, derived using the Riccati–Bernoulli sub-ODE method combined with Bäcklund transformation and conformable fractional derivatives. The obtained solutions are expressed in trigonometric, hyperbolic, and rational forms, highlighting the adaptability and efficacy of the proposed approach. To enhance the understanding of the results, the solutions are visualized using 2D representations for fractional-order variations and 3D plots for integer-type solutions, supported by detailed contour plots. The findings contribute to a deeper understanding of nonlinear wave–wave interactions and the underlying dynamics governed by fractional-order derivatives. This work underscores the significance of fractional calculus in analyzing complex wave phenomena and provides a robust framework for further exploration in nonlinear sciences and optical wave modeling.

1. Introduction

This research work begins by reviewing the existing literature on the topic and introducing the Kuralay-II system. This section provides the reader with a brief overview of the work’s goals, a summary of previous work and a guide to the manuscript’s layout.

Nonlinear partial differential equations (NPDEs) have been used in many applied sciences over the past few decades, such as atomic, particle, optical, nuclear, and biological physics [1,2]. Such equations are necessary when describing various subtle effects, due to which different nonlinear models have been created. A variety of techniques have been developed to solve numerical, analytical, and some semi-analytical solutions, including the spline scheme [3], finite difference method [4], Adomian decomposition technique [5], variational iteration [6] and several expansion techniques, such as sine–cosine expansion and the modified simple equation method [7,8]. Recently, a considerable amount of attention has been paid to work on the interactions of nonlinear dynamic systems and solitons because of their widespread utility in optics and fluid mechanics, as well as in materials science [9,10]. Powerful analytical solutions for numerous differential equations have been extended through generalized systems and algebraic constraints, and the application of fractional and perturbation models reflect the physical phenomena of waves, fluids and electromagnetic vibrations [11]. This paper adds to the literature as a development of the mathematical proficiency and generality of solutions by using the Bäcklund transformation and fractional calculus. The hope is that, by combining these techniques with previously existing ones, an enhanced approach to analysis and decision-making in the context of uncertainty and complexity will be developed [12,13,14,15,16].

It is necessary to use fractional partial differential equations (FPDEs) to describe many natural processes in biology, applied physics, chemistry, economics, etc. Some of the real applications of these theories involve the study of material and wave behaviors in such domains, which require nonlinear PDEs such as the extended Zakharov–Kuznetsov equation [17], fifth-order Lax equation [18], Fokas equation [19], Clannish Random Walker Parabolic [20], Oskolkov equation [21], Schrödinger dynamical systems [22] and dispersive long-wave equations [23]. There are some new techniques, including Lie symmetry analysis [24], the Jacobi elliptic function method [25], the Khater method [26] and the ($\psi$–$\phi$)-expansion method [27]. These are employed to find analytical soliton solutions, such as exact solutions in the form of solitons for the fractional partial differential equations (FPDEs) [28,29,30,31,32]. Searching for various explicit solutions with the help of analytical methods is essential for the study of behavior in different scientific fields, such as fluid mechanics, wave propagation and engineering applications at the basis of nonlinear partial differential equations (NPDEs). The NPDE soliton solutions are useful as they are explicit solutions, and solitons preserve their shape and velocity through time-introducing stability and order into the nonlinear system [33,34,35]. Solitons play the role of stable information bearers, which makes them important in theoretical as well as practical studies. Thus, to obtain new soliton solutions, a range of strong techniques are applied. The general methods of solving the NPDEs include the Sin-Gordon [36] and exp-function [37] approaches, the Sardar sub-equation [38], the extended direct algebraic method (EDAM) [39] and the Hirota bilinear approach [40]. These methods, which are constantly being fine-tuned, help in the development of the toolkit that is used in the analysis of new solutions.

The objective of this paper is to discuss the influence of this new derivative on the solutions of the time–space fractional Kuralay-II system by employing the Riccati–Bernoulli sub-ODE method combined with the Bäcklund transformation, which have not been adopted in previous studies. The fractional derivative represents both integer and fractional-order behaviors. It improves classical calculus by including memory effects, which are important in many scientific processes. Many fractional models exist in parallel to experimental data so that their application across the fields of physics and engineering is feasible. In particular, the present work reveals that lower fractional orders correspond to greater-magnitude derivatives with convergence as the fractional orders rise, indicating the derivative ability to track unique soliton dynamics. Our study, based on trigonometric, hyperbolic and rational functions, reveals different solutions and relations between the considered fractional NPDEs and integer ones. These methods provide a basic but effective means to obtain different solutions to specific FNLPDEs more effectively than conventional approaches. The Kuralay-II system (K-IIS) plays a critical role in understanding and modeling nonlinear wave phenomena across various physical and engineering domains. Its applications are particularly prominent in the field of nonlinear optics, where it is utilized to describe wave interactions in optical fibers, waveguides and other photonic systems. The soliton solutions derived in this study offer valuable insights into the behavior of optical pulses, making them relevant for several advanced technologies. The K-IIS has been used to model the features of solitons in optical communication systems. In this regard, the soliton solutions determined from fractional derivatives encompass higher-order dispersion and nonlinearity that cannot be described by the standard integer-order models. This is particularly advantageous in the development of high-data-capacity, long-haul communication networks, where pulse spreading and distortion effects have to be kept at their lowest.

Other novel and developing uses of the K-IIS are in investigations of nonlinear waves in metamaterials. These manmade materials possess novel characteristics, including a negative refractive index; as such, to describe wave behaviors within them, highly sophisticated models are need. It is expected that the fractional-order solutions obtained in this work can be applied to analyze energy localization and transmission in nonlinear metamaterial layers for the purpose of designing new devices for efficient wave control, e.g., cloaking systems or energy concentrator equipment. This analytically exact work offers practical tools for the armory of researchers and engineers, enabling them to analytically study nonlinear phenomena in a wide array of disciplines, from photonics and materials science to nanotechnology and fluid dynamics.

This article considers the nonlinear fractional Kuralay-II system (K-IIS), which has been discussed in [41]:

$$\begin{array}{cc}\hfill & i{D}_t^{\alpha }\left(f\right)-D_x^{\alpha }\left(D_t^{\alpha }\left(f\right)\right)-gf=0,\hfill \\ \hfill & D_x^{\alpha }\left(g\right)-2\beta \left(D_t^{\alpha }\left({\left|g\right|}^2\right)\right)=0,0<\alpha \le 1.\hfill \end{array}$$

(1)

For the study of the Kuralay-II system, while assigning the values to (f), we take $f=f(x,t)$ as a complex function and $g=g(x,t)$ as a real function and $\beta =\pm 1$. The spatial dimension is outlined by variable (x), while the temporal dimension is represented by variable (t). This time–space fractional Kuralay-II equation is useful to model complex natural processes because it contains the integer and fractional orders of derivatives. The elements of Cauchy’s fraction in the given equation improve the modeling in a wide range of fields, such as applied mathematics, physics, nonlinear optical phenomena, fluid dynamics, communication systems, etc. [42]. This fractional approach enables researchers to better understand problems across these fields and offers analysis and calculation benefits. It is the fundamental correlation between gauge and geometrical equivalence that is paramount in soliton theory, particularly in the study of the fractional Kuralay-II equation. Some new soliton solutions have been obtained through sophisticated techniques and analyses in recent research. For instance, Sagidullayeva et al. [41] used the Hirota bilinear method to show that two nonlinear models are gauge-equivalent. Faridi et al. [42] and Mathananaranjan [43] used different expansions to derive soliton, solitary wave and modulation instability solutions to some of these equations. Zafar et al. [44] and Khan et al. [45] discussed exact solutions with the help of different techniques, such as $ex{p}_a$-function and Jacobi elliptic function expansion, and found some applications in science and engineering. According to previous studies on the fractional Kuralay-II system equations, there is no work on periodic perturbation solitons (PPSs) utilizing our outlined Riccati–Bernoulli sub-ODE and Bäcklund transformation techniques. This illustrates the present research gap in the literature. To fill this gap, our study applies these methods to acquire and analyze PPSs in the fractional Kuralay-II system to gain a further understanding of soliton dynamics, periodicity and the real-world applications of this model to a wide range of nonlinear systems.

In our work, we employ conformable fractional derivatives (CFDs) [46], which are suitable tools for the modeling of fractional systems since they enrich the integer-order calculus in terms of the properties. The conformable fractional derivative is defined for a function $f\left(x\right)$ as

$$\begin{array}{c}\hfill T^{\alpha }\left(f\right)\left(x\right)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{f(x+ϵx^{1-\alpha })-f\left(x\right)}{ϵ}},0<\alpha \le 1.\end{array}$$

(2)

$$\left\{\begin{array}{c}{T}^{\alpha }(af+bg)=a{T}^{\alpha }\left(f\right)+b{T}^{\alpha }\left(g\right).\hfill \\ T^{\alpha }\left(fg\right)\left(x\right)=f\left(x\right)T^{\alpha }\left(g\right)\left(x\right)+g\left(x\right)T^{\alpha }\left(f\right)\left(x\right).\hfill \\ T^{\alpha }\left({\displaystyle \frac{f}{g}}\right)\left(x\right)={\displaystyle \frac{g\left(x\right)T^{\alpha }\left(f\right)\left(x\right)-f\left(x\right)T^{\alpha }\left(g\right)\left(x\right)}{g{\left(x\right)}^2}}.\hfill \\ T^{\alpha }\left(f\left(g\left(x\right)\right)\right)=f^{\prime }\left(g\left(x\right)\right)T^{\alpha }\left(g\right)\left(x\right).\hfill \end{array}\right.$$

(3)

Such properties enable us to obtain straightforward physical explanations and improve the effectiveness in projecting existing circumstances in various disciplines, including fluid mechanics and wave theory. In our study, the CFDs enable the continuous treatment of the periodic perturbation solitons in the fractional Kuralay-II model, which incorporates memory effects but is numerically manageable. This is essential in achieving improved accuracy for soliton solutions, as well as a deeper understanding of the model’s dynamical properties under fractional effects.

2. Methodology

Consider the nonlinear partial differential equation (PDE)

$$U\left(u,D_t^{\alpha }u,D_x^{\alpha }u,D_t^{2\alpha }u,D_x^{2\alpha }u,\dots \right)=0.$$

(4)

where U is some polynomial in $u(x,t)$ and its partial derivatives, but, more specifically, the last derivatives and nonlinear terms thereof. Our method [47,48,49] proceeds by applying the wave transformation

$$U(x,t)=u\left(\psi \right)=a{\displaystyle \frac{x^{\alpha }}{\alpha }}-b{\displaystyle \frac{t^{\alpha }}{\alpha }}.$$

(5)

with constants a and b, transforming Equation (4) into an ordinary differential equation (ODE):

$$V\left(u,{\displaystyle \frac{du}{d\psi }},{\displaystyle \frac{d^{2}u}{d{\psi }^2}},{\displaystyle \frac{d^{3}u}{d{\psi }^3}},\dots \right)=0.$$

(6)

where V is a polynomial in $u\left(\psi \right)$ and its derivatives. We assume the formal solution form for Equation (6):

$$U(x,t)=u\left(\psi \right)=\sum _{i=-m}^m{a}_i{T\left(\psi \right)}^i,$$

(7)

where the constants ($a_i$) are decided with $a_m\ne 0$ or $a_{-m}\ne 0$, where $T\left(\psi \right)$ is derived from a Bäcklund transformation:

$$T\left(\psi \right)={\displaystyle \frac{-\tau R_2+R_1\varphi \left(\psi \right)}{R_1+R_2\varphi \left(\psi \right)}}$$

(8)

where $\tau ,R_1$ and $R_2$ are constants with ($R_2\ne 0$), and where $\varphi \left(\psi \right)$ satisfies the Riccati equation:

$${\displaystyle \frac{d\varphi }{d\xi }}=\tau +\varphi {\left(\psi \right)}^2.$$

(9)

From previous studies [50], it is known that the Riccati equation has solutions

$$\begin{array}{cc}\hfill & \varphi \left(\psi \right)=\left\{\begin{array}{cc}-\sqrt{-\tau }tanh\left(\sqrt{-\tau }\psi \right),\hfill & \mathrm{as}\tau <0,\hfill \\ -\sqrt{-\tau }coth\left(\sqrt{-\tau }\psi \right),\hfill & \mathrm{as}\tau <0,\hfill \end{array}\right.\hfill \\ \hfill & \varphi \left(\psi \right)=-{\displaystyle \frac{1}{\psi }},\mathrm{as}\tau =0,\hfill \\ \hfill & \varphi \left(\psi \right)=\left\{\begin{array}{cc}\sqrt{\tau }tan\left(\sqrt{\tau }\psi \right),\hfill & \mathrm{as}\tau >0,\hfill \\ -\sqrt{\tau }cot\left(\sqrt{\tau }\psi \right),\hfill & \mathrm{as}\tau >0.\hfill \end{array}\right.\hfill \end{array}$$

(10)

In the case of finding the integer (m) for Equation (7), we use a compromise between the highest derivatives and the nonlinear terms in the transformed ODE. Defining the degree of $u\left(\psi \right)$ as $D\left[u\right(\psi \left)\right]=m$ allows us to compute the degrees of related expressions, such as

$$\begin{array}{cc}\hfill & D\left[{\displaystyle \frac{d^{q}u}{d{\psi }^q}}\right]=m+q\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & D\left[u^p{\left({\displaystyle \frac{d^{q}u}{d{\psi }^q}}\right)}^s\right]=mp+s(q+m)\hfill \end{array}$$

(12)

Since the ODE is satisfied by both our assumptions and the Riccati solutions, we replace Equations (7) and (9) with Equation (4) to obtain an algebraic system, which is soluble, using Maple for the values of $\tau$, $\delta$, $\mu$, $\lambda$ and $a_i$. This gives exact traveling wave solutions for the original PDE as it yields, in general, solutions for various well-known traveling wave equations. In case $R_2=0$, this particular approach will be reduced to the modified extended tanh-function method.

3. Execution of the Problem

In this paper, we use the proposed Riccati–Bernoulli sub-ODE method to find the optical soliton solutions of Equation (1). The process begins with the introduction of a complex wave transformation formulated as follows:

$$\begin{array}{cc}\hfill & f(x,t)=e^{i\psi }F\left(\zeta \right),g(x,t)=G\left(\zeta \right),\hfill \\ \hfill & \zeta ={\displaystyle \frac{\lambda x^{\alpha }}{\alpha }}+{\displaystyle \frac{\delta t^{\alpha }}{\alpha }},\psi ={\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\hfill \end{array}$$

(13)

Upon substituting Equation (13) into Equation (1) and separating the imaginary and real components, we obtain

$$\begin{array}{cc}\hfill & (\delta -\rho \delta -\mu \zeta )i{\displaystyle \frac{dF}{d\zeta }}-\delta \zeta {\displaystyle \frac{d^{2}F}{d{\zeta }^2}}+(\rho \mu -G-\beta )F=0,\hfill \\ \hfill & \zeta {\displaystyle \frac{dG}{d\zeta }}-4\beta \delta F{\displaystyle \frac{dF}{d\zeta }}=0,F=F\left(\zeta \right),G=G\left(\zeta \right).\hfill \end{array}$$

(14)

Integrating the second part of Equation (14) with respect to y and assuming that the constant of integration is zero, we find

$$\begin{array}{c}\hfill G={\displaystyle \frac{2\beta \delta F^2}{\zeta }}.\end{array}$$

(15)

Substituting Equation (15) back into Equation (14) simplifies the real part of the system, reducing it to a single nonlinear ordinary differential equation (NODE):

$$\begin{array}{c}\hfill (\rho -1)\mu F-\zeta \delta {\displaystyle \frac{d^{2}F}{d{\zeta }^2}}-{\displaystyle \frac{2\beta \delta F^3}{\zeta }}=0.\end{array}$$

(16)

where $\rho ={\displaystyle \frac{\delta -\mu \zeta }{\delta }}$ is the constraint condition of the imaginary part. With this reformulation, we are then able to use the Riccati–Bernoulli sub-ODE method in conjunction with the Bäcklund transformation to systematically break down and analyze the nonlinearity of the system, which is important in understanding the spatial–temporal behaviours of optical solitons in a fractional-order context. This methodology sheds light on the effects that fractional parameters have on soliton behavior for use in nonlinear optics and other systems where wave–wave interactions occur. If the balancing procedure is implemented between ${\displaystyle \frac{d^{2}F}{d{\zeta }^2}}$ and ($F^3$), then ($m=1$) is obtained. By substituting Equations (7) and (9) into Equation (16), and then equating the coefficients of various powers of $\varphi \left(\psi \right)$ to zero, a set of algebraic equations results. It helps to reach a conclusion regarding the most suitable unknown parameters for a given system, which provides the exact solutions for the same. The equations obtained in the course of this procedure form a system of algebraic equations, which can be easily solved by different programs for example Maple, Mathematica, MATLAB or analogous. This enables the identification of multiple distinct solution cases, outlined as follows:

$$\begin{array}{cc}& {\varphi }^0:6\beta \delta {a_{-1}}^2{b_2}^6{a}_0\tau =0,\hfill \\ & {\varphi }^1:-6\beta \delta {a_{-1}}^2{b_2}^6{a}_1{\tau }^2-6\beta \delta a_{-1}{b_2}^6{a_0}^2{\tau }^2-\mu a_{-1}{b_2}^6\lambda {\tau }^2-2\delta {\lambda }^2{a}_{-1}{b_2}^6{\tau }^3+\mu \rho a_{-1}{b_2}^6\lambda {\tau }^2,\hfill \\ & {\varphi }^2:-2\delta {\lambda }^2{a}_{-1}{b_2}^6{\tau }^2-2\beta \delta {a_{-1}}^3{b_2}^6=0,\hfill \\ & {\varphi }^3:\mu \rho a_1{\tau }^4{b_2}^6\lambda -6\beta \delta {a_0}^2{a}_1{\tau }^4{b_2}^6-6\beta \delta a_{-1}{b_2}^6{a_1}^2{\tau }^4-\mu a_1{\tau }^4{b_2}^6\lambda -2\delta {\lambda }^2{a}_1{b_2}^6{\tau }^5=0,\hfill \\ & {\varphi }^4:6\beta \delta a_0{a_1}^2{\tau }^5{b_2}^6=0,\hfill \\ & {\varphi }^5:-\mu \rho a_0\lambda {\tau }^3{b_2}^6+12\beta \delta a_{-1}{b_2}^6{a}_0{a}_1{\tau }^3+2\beta \delta {a_0}^3{\tau }^3{b_2}^6+\mu a_0\lambda {\tau }^3{b_2}^6=0,\hfill \\ & {\varphi }^6:-2\delta {\lambda }^2{a}_1{b_2}^6{\tau }^6-2\beta \delta {a_1}^3{\tau }^6{b_2}^6=0.\hfill \end{array}$$

(17)

• Case 1:

$$\begin{array}{c}\hfill a_0=0,a_1=a_1,a_{-1}=0,\mu =\mu ,\delta =\delta ,\beta =-{\displaystyle \frac{{\lambda }^2}{{a_1}^2}},\tau =1/2{\displaystyle \frac{\mu \left(\rho -1\right)}{\lambda \delta }},\lambda =\lambda .\end{array}$$

(18)

• Case 2:

$$\begin{array}{c}\hfill a_0=0,a_1=a_1,a_{-1}=a_{-1},\mu =\mu ,\delta =-1/4{\displaystyle \frac{a_1\mu \left(\rho -1\right)}{a_{-1}\lambda }},\beta =-{\displaystyle \frac{{\lambda }^2}{{a_1}^2}},\tau ={\displaystyle \frac{a_{-1}}{a_1}},\lambda =\lambda .\end{array}$$

(19)

In this regard, the two cases obtained using the algebraic system described above may be used to identify families of solutions with regard to $\tau$. These solution families show different behaviors of the system in different cases. The emergent solution behaviors reveal the temporal dynamics of the system, as well as the physical manifestations and mechanisms under a variety of fractional-order controls and initial states. Moreover, these families of solutions include all the interconnections distinguished in the system, while revealing the impacts of different $\tau$ on soliton behavior and stability, and it has significance for fluid dynamics and nonlinear waves.

From here, we concentrate on solving Case 1 by inserting the obtained values of ($\tau$) and reveal a set of unique solutions for Equation (1). The solutions obtained here have periodic perturbations in solitons, which suggests that the soliton profile oscillates temporally within this setting. All such fluctuations indicate that the solitons undergo rhythmic disturbances that affect their amplitude as well as stability. This is important for the analysis of the time dependence of the behavior and possible resonances in the system under changes in conditions.

Solution set 1: ($\tau <0$).

$$f_1(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-1/2b_1\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tanh\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-1/2b_2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tanh\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.$$

(20)

or

$$\begin{array}{c}\hfill f_2(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-1/2b_1\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}coth\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-1/2b_2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}coth\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.\end{array}$$

(21)

Solution set 2: ($\tau >0$).

$$\begin{array}{c}\hfill f_3(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }+1/2b_1\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tan\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1+1/2b_2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tan\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.\end{array}$$

(22)

or

$$\begin{array}{c}\hfill f_4(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-1/2b_1\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}cot\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-1/2b_2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}cot\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.\end{array}$$

(23)

Solution set 3: ($\tau =0$).

$$\begin{array}{c}\hfill f_5(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-b_1{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}{\left(b_1-b_2{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}}·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}\end{array}$$

(24)

Here, we show the associated solutions for (Case 2) and ($\tau <0$).

$$\begin{array}{c}\hfill f_6(x,t)=\left[{\displaystyle \frac{a_{-1}\left(b_1-b_2\sqrt{-\frac{a_{-1}}{a_1}}tanh\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{-\frac{a_{-1}}{a_1}}tanh\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}+{\displaystyle \frac{a_1\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{-\frac{a_{-1}}{a_1}}tanh\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-b_2\sqrt{-\frac{a_{-1}}{a_1}}tanh\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}\right]\\ \hfill ·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}\end{array}$$

(25)

or

$$\begin{array}{c}\hfill f_7(x,t)=\left[{\displaystyle \frac{a_{-1}\left(b_1-b_2\sqrt{-\frac{a_{-1}}{a_1}}coth\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{-\frac{a_{-1}}{a_1}}coth\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}+{\displaystyle \frac{a_1\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{-\frac{a_{-1}}{a_1}}coth\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-b_2\sqrt{-\frac{a_{-1}}{a_1}}coth\left(\sqrt{-\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}\right]\\ \hfill ·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}\end{array}$$

(26)

Solution set 4: ($\tau >0$).

$$f_8(x,t)=\left[{\displaystyle \frac{a_{-1}\left(b_1+b_2\sqrt{\frac{a_{-1}}{a_1}}tan\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(-\frac{a_{-1}{b}_2}{a_1}+b_1\sqrt{\frac{a_{-1}}{a_1}}tan\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}+{\displaystyle \frac{a_1\left(-\frac{a_{-1}{b}_2}{a_1}+b_1\sqrt{\frac{a_{-1}}{a_1}}tan\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1+b_2\sqrt{\frac{a_{-1}}{a_1}}tan\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}\right]·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}$$

(27)

or

$$f_9(x,t)=\left[{\displaystyle \frac{a_{-1}\left(b_1-b_2\sqrt{\frac{a_{-1}}{a_1}}cot\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{\frac{a_{-1}}{a_1}}cot\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}+{\displaystyle \frac{a_1\left(-\frac{a_{-1}{b}_2}{a_1}-b_1\sqrt{\frac{a_{-1}}{a_1}}cot\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-b_2\sqrt{\frac{a_{-1}}{a_1}}cot\left(\sqrt{\frac{a_{-1}}{a_1}}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}\right]·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}$$

(28)

Solution set 5: ($\tau =0$).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f_{10}(x,t)=\left[{\displaystyle \frac{a_{-1}\left(b_1-b_2{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}{\left(-\frac{a_{-1}{b}_2}{a_1}-b_1{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}}+{\displaystyle \frac{a_1\left(-\frac{a_{-1}{b}_2}{a_1}-b_1{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}{\left(b_1-b_2{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}}\right]\\ \hfill ·e^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}\end{array}\end{array}$$

(29)

4. Results and Discussion

In this paper, a new approach to solving space–time fractional nonlinear PDEs in nuclear physics, particle physics and fluid dynamics is proposed, employing the Bäcklund transformation and the Riccati–Bernoulli sub-ODE method to develop a closed-form solution in terms of periodic perturbations. In this work, this approach has been utilized to perform a direct and simplified transformation from PDEs to ODEs with the aim of obtaining clear and exact hyperbolic, rational and trigonometric solitons and analyzing their periodic characteristics. This understanding of periodic disturbances in a space–time fractional scheme not only enriches soliton theory but also gives adequate grounds for stability and wave analysis, with potential areas of application including topological solitons and materials science.

Table 1. Comparison of the proposed method with the alternative approach, the extended exp-function method [51].

Case I: $\tau <0$    Present method

$f(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-1/2b_1\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tanh\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1-1/2b_2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tanh\left(1/2\sqrt{-2\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}{e}^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.$

Case I: $\rho \sigma <0$    Extended exp-function method

$p(x,t)=\pm c\sqrt{{\displaystyle \frac{\rho \sigma }{\mu }}}\times sgn\left(\rho \right)tanh\left(\sqrt{-\rho \sigma }\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(c{x}^{\alpha }+d{t}^{\alpha }\right)+A\right)\right)e^{i\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(a{x}^{\alpha }+b{t}^{\alpha }\right)\right)}$.

Case II: $\tau >0$ Present method

$f(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }+1/2b_1\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tan\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}{\left(b_1+1/2b_2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}tan\left(1/2\sqrt{2}\sqrt{\frac{\mu \left(\rho -1\right)}{\delta \lambda }}\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)\right)\right)}}{e}^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.$

Case II: $\rho \sigma >0$    Extended exp-function method

$p(x,t)=\pm ic\sqrt{{\displaystyle \frac{\rho \sigma }{\mu }}}\times sgn\left(\rho \right)cot\left(\sqrt{\rho \sigma }\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(c{x}^a+d{t}^a\right)+A\right)\right)e^{i\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(a{x}^{\alpha }+b{t}^{\alpha }\right)\right)}$.

Case III: $\tau =0$    Present method

$f(x,t)={\displaystyle \frac{a_1\left(-1/2\frac{\mu \left(\rho -1\right)b_2}{\delta \lambda }-b_1{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}{\left(b_1-b_2{\left(\frac{\lambda x^{\alpha }}{\alpha }+\frac{\delta t^{\alpha }}{\alpha }\right)}^{-1}\right)}}{e}^{i\left({\displaystyle \frac{\rho x^{\alpha }}{\alpha }}+{\displaystyle \frac{\mu t^{\alpha }}{\alpha }}\right)}.$

Case III: $\rho =0,\sigma >0$    Extended exp-function method

$p(x,t)=\pm ic\sqrt{{\displaystyle \frac{1}{\mu }}}{\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(c{x}^a+d{t}^{\alpha }\right)+A\right)}^{-1}{e}^{i\left({\displaystyle \frac{\Gamma (1+{\rm Y})}{\alpha }}\left(a{x}^{\alpha }+b{t}^a\right)\right)}$.

, comparison of the proposed method with the alternative approach, the extended exp-function method [51].

As shown in the study on the fractional Kuralay-II system—see Figure 1, Figure 2, Figure 3 and Figure 4—a periodic perturbation is displayed with the $3D$ integer-order and the fractional-order parameter ($\alpha$), as shown in the $2D$ plots of the real and imaginary components. Moreover, the contour representations in all figures confirm the presence of fractal patterns, which increases the richness of the analysis. The following is a detailed view of each figure.

Figure 1: For fractional values $0.9,0.8,$ and $0.7$, both the real and imaginary parts show a reduction in amplitude as the fractional parameter decreases, gradually moving toward convergence. The integer-order plot shows the full periodic perturbation in three dimensions, highlighting the system’s unaltered amplitude and wavelength properties. In the contour representation, fractal patterns emerge, illustrating the self-similarity within the periodic perturbations. This fractal aspect implies potential applications in fractal-based optical systems, where understanding these structures can lead to innovations in signal integrity.

Figure 2: As in the case of Figure 1, but with a longer wavelength. This larger wavelength shows how the fractional order affects the spatial spread over long extents, but the real and imaginary parts are slightly more convergent. The longer wavelength is more apparent in the integer-order plot because it provides a spatial reference for the fractional wavelength’s impact. The fractal contour patterns continue here with relatively more distant periodical peaks; these are useful in wave transmission systems in which wavelength modifications are vital.

Figure 3: In this depiction, the fractional-order solutions converge more quickly and stabilize the real and imaginary parts. This accelerated rate of convergence indicates that it may be possible to have controlled damping in engineered systems to achieve rapid stabilization. The integer-order three-dimensional representation is characterized by steady oscillations and fractal patterns pointing to a faster decrease in amplitude. The fractal structures used here describe the fast convergence behavior, which is useful in areas requiring fast stabilization, such as resonant damping in structural engineering.

Figure 4: Unlike the other figures, there is no superposition; the periodically interrupted oscillation is constant at any fractional values of the parameter. Both the real and imaginary components exhibit oscillation, with the constant amplitudes signifying the capability for energy storage in long-term periods. Moreover, the $3D$ representation of the integer order exhibits persistent periodicity, which is useful in cases when regular oscillatory behavior is needed, such as in wave guides or optical lattices. The contour plots of the displays are self-similar throughout the periodic wave, which is useful in fields where stable fractal wave structures are employed for unbroken signal transmission, such as the fractal antennas used in telecommunications. Figure 5, Visualization of the intensity of $F_1(x,t)$ of sub-figures (a), (b) and (c). Figure 6, Visualization of the intensity of $F_3(x,t)$. Figure 7, Visualization of the intensity of $F_5(x,t)$ of sub-figures (a), (b) and (c).

To strengthen our results, we include figures that represent the intensity of the wave solutions in terms of the squared modulus of the analytic solution. These visualizations give some information concerning the localization of the energy, as well as the general behavior of the wave solutions, which is critically important in the analysis of the nonlinear optics. This addition provides better robustness, making our findings consistent with the norms employed in nonlinear optics and similar disciplines.

5. Conclusions

In this study, we have investigated the periodic perturbation feature of solutions of the Kuralay-II system with fractional derivatives using the Ricatti–Bernoulli sub-ODE method and Bäcklund transformation. Based on the $2D$ and $3D$ plots of the real and imaginary parts of integer and fractional-order values, our results show the different behaviors of the waves in terms of convergence, amplitude modulation and changes in wavelength. Significantly, the features that appear in the contour plots include fractals, which give more depth to the self-similar features present in periodic solutions. Thus, this work proves that the fractional Kuralay-II system can flexibly approximate various wave dynamics, which is valuable since the fractional-order approach is effective in simulating both the periods and convergence of waveforms. Altogether, the fractional Kuralay-II system offers a suitable context for the modeling and analysis of wave phenomena, as well as for related areas of optics, signal processing and engineering, where the control of wave behavior, stabilities and energy losses is essential. Further developments of this work may investigate the complex relationships between the fractional parameters and waveforms and the deeper utilization of this model in numerous scientific and engineering fields.