Analyzing the dynamical sensitivity and soliton solutions of time-fractional Schrödinger model with Beta derivative

In physical domains, Beta derivatives are necessary to comprehend wave propagation across various nonlinear models. In this research work, the modified Sardar sub-equation approach is employed to find the soliton solutions of (1+1)-dimensional time-fractional coupled nonlinear Schrödinger model with Beta fractional derivative. These models are fundamental in real-world applications such as control systems, processing of signals, and fiber optic networks. By using this strategy, we are able to obtain various unique optical solutions, including combo, dark, bright, periodic, singular, and rational wave solutions. In addition, We address the sensitivity analysis of the proposed model to investigate the truth that it is extremely sensitive. These studies are novel and have not been performed before in relation to the nonlinear dynamic features of these solutions. We show these behaviors in 2-D, contour 3-D structures across the associated physical characteristics. Our results demonstrate that the proposed approach offers useful results for producing solutions of nonlinear fractional models in application of mathematics and wave propagation in fiber optics.


Mathematical model
The time FCNLS model in (1+1) dimension containing a fractional derivation (FD) of beta is as follows 30 (1) ( ). (3) www.nature.com/scientificreports/ In Beta FD, D ς t and D 2ς x are real valued functions and U and R are complex functions.Nonlinear behavior in time FCNLS model causes the effects which extend over a simple linear composition of its parameter aspects.This nonlinearity enables exciting phenomena such as the generation of solitons, self-destructive, and change of energy among connected components.The study of the (1+1)-dimensional time FCNLS model with beta derivatives has gained increasing significance due to its numerous applications in various domains.Travelling waves in fractal media have been described via analytical solutions for a class of FCNLS models.The FCNLS model has been transformed into ordinary differential equations using new conformable fractional derivative techniques, which has made it easier to derive precise traveling wave solutions.Fractional dual-function and fractional Riccati methods have been used to find vector photonic soliton and periodic solutions for the FCNLS model.Fractional space-time derivatives have been the focus of investigations on the FCNLS model and new explosives 31,32 .Our paper proposes novel techniques to handle this complex problem using the MSSE approach and obtain the optical soliton solutions, time series, and sensitivity analysis 33 .This study has tremendous implications for engineering and scientific research since it provides insights into complex system behaviors and aids in the development of appropriate control systems.This research offer new avenues for research, especially because they have never been applied to the time FCNLS model.Our work follows a more general strategy, spanning a wide range of optical solutions and concentrating on specific solution types.

Methodology of the MSSE approach
The modified Sardar sub-equation (MSSE) approach expands on the original Sardar sub-equation approach by incorporating additional variables and scenarios into the ansatz for solving nonlinear problems.This approach has been successfully applied to solve NLPDEs in many different areas of mathematics and science.The general form of NLPDEs is Step-i: Utilizing the complex wave transformation into Eq.( 4), we obtain Thus, the nonlinear ordinary differential equations (NLODEs) is achieved as Step 2. According to the approach, the general solution of Eq. ( 8) is described in the following form where M = M(η) assures where ω 0 = 1 , ω 1 and ω 2 = 0 are integers.Compute the constants F 0 and F 1 .Moreover, F j is invertible, thus it can be zero.The values of J can be obtained by using balance principle.The cases to Eq. ( 10) are as follows. Case-1:

Case-7:
• Let ω 0 = 0, ω 1 = 0 and ω 2 > 0 , then Step 3. Put Eq. ( 9) into Eq.( 8) and by using Eq. ( 10), the polynomial can be obtained as a power of L(η) .Step 4. Assemble the similar parameters of L(η) and equating them to zero, we can obtain the algebraic system for F 0 , F j ( j = 1, 2, 3, ... ).Step 5. Finally, apply the Mathematica Software to the algebraic systems of equations to obtain the coefficients values.Putting these parameter values to Eq. ( 8), we get the solution of Eqs.(3, 4 and 5).The MSSE approach is a helpful tool for obtaining the precise results to NLPDEs, such as the (1+1)-dimensional FCNLS model.This method requires assuming an ansatz for results in terms of additional variables and a unique function, and then solving an algebraic system of equations to obtain the unknown constants.

Mathematical analysis
This part concentrates on implementing our suggested approach to validate its effectiveness, performance, and reliability.Consequently, we obtain a soliton solution for the time-dimensional (1+1) FCNLS model.The Eq. ( 7) containing the complex transformation is employed.The Eq. ( 3) is now utilized to convert Eqs.(3), (4) and (5) into NLODEs.Consequently, the real and imaginary parts of NLODEs yields and solve Eq. ( 33), we get putting Eq. ( 34), into Eq.( 32), we get Inserting Eq. (7) into Eqs.(4 and 5), and then integrate, we get the following Eqs.

Dynamical system
A system of dynamic is applied to describe the temporal dependency of a location of the point within its connecting area 34 .It is a collection of criteria that outline how parameters shift through time, the sensitivity of the concerning model, and how a system evolves.Dynamic systems are widely used in many fields, such as mathematics, biological sciences, chemistry, science and engineering, and financial studies.Population dynamics, chemical reactions, engineering problems, and the Schrödinger model are among the applications for these systems.Complex instances that predict the effects of changes in a range of sectors necessitate a deep understanding of dynamic applications and structures.The Eq. ( 38) can be turned into a dynamical framework after utilizing a particular modification.Now, consider After applying the aforementioned transformation to Eq. ( 104), we obtained the dynamical system that follows We can obtain the sensitivity analysis of the concerning framework utilizing the dynamical system of Eq. ( 105) by applying varied time-variant and initial conditions.

Sensitivity analysis
Sensitivity analysis is a mathematical approach to assessing the effect of alternations in a framework of variables on its output.It is crucial to understand the capacity and reliability of dynamic structures.This analysis is commonly applied to investigate the changes in variables or configuration that affect the performance of systems in several types of disciplines, including energy, ecological structures, and dynamical framework 35 .The graphical representations of sensitivity analysis under appropriate parameter values and initial conditions are shown in Figs.(14-17). (99)

Results and discussions
In this section, we compare some of our most current research findings with previous published study.Shakeel et al. 30 employed the exponential rational approach to explore the results of time FCNLS model involving Beta derivative.In our present work, we consider (1+1)-dimensional time FCNLS model including Beta derivative and apply the MSSE approach to obtain dark, singular, periodic and rational solutions.This work presents a new technique to investigate sensitivity in model dynamics, performing time series analysis, and obtaining the optical soliton solutions.These methods work well, are simple to use, and can be applied to a variety of complex systems.Earlier research, on the other hand, was mainly concerned with determining optical soliton solutions and investigating sensitivity in model behavior.We build on this in our research by adding time series analysis, which offers a thorough comprehension of the dynamic behavior of the model.In addition, our findings provide new perspectives on how to use MSSE to investigate sensitivity and obtain soliton solutions, which advances the field of nonlinear dynamics studies.Localised areas of a wave's lower intensity are represented by dark solitons.
In physical systems such as optical fibers, they correspond to regions of minimum light intensity, frequently as a result of dispersion being counteracted by nonlinear processes.Localized intensified patches within a wave are the defining feature of bright solitons.Within optical systems, they represent regions of high light intensity, usually due to nonlinear effects counteracting dispersion.Within a wave, sudden changes or areas of severe behavior are indicated by singular soliton solutions.The study of wave dynamics, including electromagnetic wave propagation, depends on these solutions, which can be found in many different physical processes.Insights into wave behavior that can be characterized by straightforward mathematical relationships are provided by rational soliton solutions, which are wave patterns controlled by rational functions.The framework of obtained solitons is depicted in Figs.A single wave with singular soliton solutions shows that derivatives are discontinuous.Compactions and peakons having peaks with discontinuous first derivatives, are two examples.Periodic solutions are very important in various branches of technology because they occur again over time.Rational approaches are very beneficial in mathematics subjects including geometry, calculus, and numerical methods.These methods help in pattern recognition, connecting various sorts of solutions, and offering insights on equation structures.This technique may be constrained by its limited applicability to particular equation types or issue domains.The efficiency of the approach depends on constraint relations on parameters, which aren't always easily met or appropriate in every situation.Controlling the method's complexity, especially when working with large equations or systems, may provide difficulties and reduce its effectiveness.Although the method yields analytical answers, it might not always provide great precision, particularly for highly nonlinear or complicated systems, which could result in errors.Even with these benefits, there might still be opportunities for algorithmic refinements or greater generalizability to a larger class of equations and issues.

Conclusion remarks
This work explores a (1+1)-dimensional temporal FCNLS model for fibre optic wave analysis that includes Beta fractional derivatives.We extract soliton solutions and perform a qualitative model evaluation using the MSSE approach.The solutions have been found in single, periodic, combination, dark, and rational solutions.Using sensitivity analysis, we investigate the sensitivity of the dynamical system and expose its dependency on several physical parameters with novel insights.These techniques provide a dynamic mathematical tool for solving a variety of nonlinear wave difficulties in mathematical physics, engineering, fibre optic waves, and other nonlinear domains.These results may be important for comprehending how fibre optic waves spread in oceanography.