
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38594393
58796
10.1038/s41598-024-58796-z
Article
Analyzing the dynamical sensitivity and soliton solutions of time-fractional Schrödinger model with Beta derivative
Nadeem Muhammad 1
Liu Fenglian zz2105@ynufe.edu.cn

2
Alsayaad Yahya yahyaalsayyad2022@hoduniv.net.ye

3
1 https://ror.org/02ad7ap24 grid.452648.9 0000 0004 1762 8988 School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 China
2 https://ror.org/04rhev598 grid.464506.5 0000 0000 8789 406X Institute of Land & Resources and Sustainable Development, Yunnan University of Finance and Economics, Kunming, 650221 China
3 https://ror.org/05fkpm735 grid.444907.a Department of Physics, Hodeidah University, Al-Hudaydah, Yemen
9 4 2024
9 4 2024
2024
14 830129 1 2024
3 4 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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.

Keywords

Nonlinear time-fractional coupled Schrödinger model
Modified Sardar sub-equation approach
Beta derivative
Soliton solutions
Sensitivity analysis
Subject terms

Applied optics
Optical physics
issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Fractional calculus (FC) is a field of mathematics that focuses on non-integer order derivatives and integrals. In recent years, various applications of FC have increased in the fields of physics, engineering, and applied mathematics. Numerous scholars have explored new theories and applications, like multiplicative fractional calculus and fuzzy logic to develop the models for real-world problems1,2. Novel kinds of inequalities are presented by FC and its applications with non-conformable fractional integrals. Numerous concepts have reported, including Riemann-Liouville, Atangana-Baleanu, Caputo-Fabrizio and conformable derivatives3–5. These fractional derivatives have applications in a wide range of scientific and technical disciplines. Wave motion in dispersive objects, viscoelastic material comprehension, fractal feature processing in signals, modeling biological systems with anomalous diffusion, and electromagnetic system understanding are some of its uses. They serve as fundamental concepts in the area of fractional calculus as well. When combined, these fractional derivatives provide an adaptable framework for explaining and understanding complex real-world phenomena. This versatility has led to advancements in several branches of signal processing, physics, biology, and materials research. Their significance originates from their capacity to describe systems with fractional-order effects, rendering them valuable tools for comprehending and addressing an extensive array of scientific and technical problems6,7.

A class of mathematical models known as nonlinear partial differential equations (NLPDEs) is needed to explain a broad range of natural events in quantum physics and wave propagation8,9. Beyond their mathematical elegance, NLPDEs have theoretical relevance because they provide a solid framework for modeling complex framework, mechanisms, and transfers in daily life challenges. Natural processes frequently exhibit nonlinear behavior; examples include soliton production, shock waves, and pattern self-organization. Because of their unpredictability, these occurrences call for intricate mathematical models that can depict the intricate relationships between numerous variables10,11.

A soliton is a specific type of solitary wave that acts as a particle and keeps its individuality if it interacts with another soliton. It is necessary to comprehend these individual waves to comprehend the dynamics of waves. Typically, a single soliton solution is referred to as a “solitary wave”. Solitons offer stable solutions for NLPDEs if the impacts of scattering and nonlinearity are completely balanced12. The concept of solitons has grown to be an exciting field of study and played a major impact on the recent developments in the telecom sector. Fibre optics has demonstrated the effectiveness of solitons in transmitting digital messages over vast distances without dispersing13,14. Various analytical and numerical techniques have been used for the solutions of nonlinear models, such as; the khater technique15, the unified technique16, the G′G technique17, the Hirota bilinear technique18, the extended tanh-function scheme19, the F-expansion technique20, the modified simple equation method21, the homotopy perturbation technique22, the modified variational iteration technique23, the sine-Gordon expansion technique24 and the direct algebraic technique25 and so on26–28.

In this paper, we examine innovative soliton solutions and conduct sensitivity analysis to validate the sensitivity of the proposed model. In this work, the use of Beta derivatives to time-fractional coupled nonlinear Schrödinger (FCNLS) model is innovative since it allows for an additional complete analysis of solution spaces and discovers the previous unknown solution of frameworks. This study is designed as: Section “Beta fractional derivative”, discussed an overview of Beta fractional derivative and a physical significance of time FCNLS model. Section “Methodology of the MSSE approach” explains the idea of modified Sardar sub-equation (MSSE) approach. Section “Mathematical analysis” describes the extraction of soliton solutions. In Section “Dynamical system”, we discuss the features of dynamical system of the proposed model with sensitivity analysis. The results and discussions are presented in Section “Results and discussions”. The conclusion remarks of this study are discussed in Section “Conclusion remarks”.

Beta fractional derivative

Recently, the concept of the Beta fractional derivative (FD) has been proposed by Atangana et al29., indicating an important development in study of mathematical derivatives. In particular, Beta FD offer greater capability in accurately predicting real-time phenomena compared to standard derivative. The main advantage of FD is its non-locality where it shows the impact of distant elements on the behavior of a system and adds fundamental value. Numerous fields, including dielectric polarization, viscoelasticity, electrical chemistry, processing of images, and magnetic systems, make extensive use of beta FD. Most of them appear in physics and engineering such as robotics, heat and mass transfer, biotechnology, and wave theory.

Definition 2.1

The Beta FD of g with order ς∈(0,1] is expressed as1 0QDtςg(t)=limγ→0g(t+γ(t+1Γ(ς)))-g(t)γ,g:(0,∞)→R.

In the open interval (0,c),c>0, g is ς-differentiable, and limt→0+(0QDtςg(t)). Then2 0QDtςg(0)=limt→0+(0QDtςg(t)).

Theorem 2.1

Let g is continuous at t0, when g:(0,∞)→R is ς-differentiable for t0>0, where ς∈(0,1].

Theorem 2.2

If 0<ς≤1, a,  b ∈ R, g,u,ς-differentiable, at a point t>0. Then 0QDtς(v1g+v2u)=v10QDtς(g)+v20QDtς(u), v1,v2∈R.

0QDtς(ψ)=0, in which ψ is constant.

0QDtς(gu)=g0QDtς(u)+u0QDtς(g).

0QDtς(gu)=u0QDtς(g)-g0QDtς(u)u2.

Assuming γ=(t+1Γ(ς))ς-1f,f→0 when γ→0, 0QDtς(g(t))=(t+1Γ(ς))ς-1dgdt, with Φ=l1ς(t+1Γ(ς))ς), as l1 is constant,

0QDtς(g(t)u(x))=ldg(t)dt.

Mathematical model

The time FCNLS model in (1+1) dimension containing a fractional derivation (FD) of beta is as follows303 DxDtςV=DxxV+21-α2|V|2V+V(R-U),

4 DtςR=-Dtς(|V|2)1+α+(1+α)DxR,

5 DtςU=-Dtς(|V|2)1+α+(1-α)DxU.

In Beta FD, Dtς and Dx2ς 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 explosives31,32. Our paper proposes novel techniques to handle this complex problem using the MSSE approach and obtain the optical soliton solutions, time series, and sensitivity analysis33. 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 is6 W(V,DxςV,Dx2ςV,DtςV,Dx2ςV,...)=0.

Step-i: Utilizing the complex wave transformation into Eq. (4), we obtain7 V(x,t)=M(η)eiθ,R(x,t)=S(η),U(x,t)=N(η)η=ax+bt+1Γ(ς)ςς,θ=cx+lt+1Γ(ς)ςς,

Thus, the nonlinear ordinary differential equations (NLODEs) is achieved as8 Q(M,M′,M′′,...)=0.

Step 2. According to the approach, the general solution of Eq. (8) is described in the following form9 M(η)=F0+∑j=1JFjLj(η),Fj≠0,

where M=M(η) assures10 L′(η)2=ω2L(η)4+ω1L(η)2+ω0,

where ω0≠1, ω1 and ω2≠0 are integers. Compute the constants F0 and F1. Moreover, Fj 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: If ω0=0,ω1>0andω2≠0, then11 L1(η)=-ω1ω2sechω1(η+τ).

If ω0=0,ω1>0andω2≠0, then12 L2(η)=ω1ω2cschω1(η+τ).

Case-2:For constants k1andk2, let ω0=0,ω1>0 and ω2=+4k1k2, then13 L3(η)=4k1ω14k12-ω2sinhω1(η+τ)+4k12-ω2coshω1(η+τ).

Case-3: For constants E1andE2, let ω0=ω124ω2,ω1<0andω2>0, then14 L4(η)=-ω12ω2tanh-ω12(η+τ).

For constants E1andE2, let ω0=ω124ω2,ω1<0andω2>0, then15 L5(η)=-ω12ω2coth-ω12(η+τ).

For constants E1andE2, let ω0=ω124ω2,ω1<0andω2>0, then16 L6(η)=-ω12ω2tanh-ω12(η+τ)+isech-2ω1(η+τ).

For constants E1andE2, let ω0=ω124ω2,ω1<0andω2>0, then17 L7(η)=-ω18ω2tanh-ω18(η+τ)+coth-ω18(η+τ).

For constants E1andE2, let ω0=ω124ω2,ω1<0andω2>0, then18 L8(η)=-ω12ω2E12+E22-e1cosh-2ω1(η+Ψ)E1sinh-2ω1(η+Ψ)+E2,

19 L9(η)=-ω12ω2cosh-2ω1(η+τ)sinh-2ω1(η+τ)+i.

Case-4: Let ω0=0,ω1<0andω2≠0, then20 L10(η)=-ω1ω2sec-ω1(η+τ).

Let ω0=0,ω1<0andω2≠0, then21 L11(η)=-ω1ω2csc-ω1(η+τ).

Case-5: Let ω0=ω124ω2,ω1>0 and ω2>0 and  E12-E22>0, then22 L12(η)=-ω12ω2tanω12(η+τ).

Let ω0=ω124ω2,ω1>0 and ω2>0 and  E12-E22>0, then23 L13(η)=--ω12ω2cotω12(η+τ).

Let ω0=ω124ω2,ω1>0 and ω2>0 and  E12-E22>0, then24 L14(η)=--ω12ω2tan2ω1(η+τ)-sec2ω1(η+τ).

Let ω0=ω124ω2,ω1>0 and ω2>0 and  E12-E22>0, then25 L15(η)=-ω18ω2tanω18(η+τ)-cotω18(η+τ).

Let ω0=ω124ω2,ω1>0 and ω2>0 and  E12-E22>0, then26 L16(η)=-ω12ω2E12-E22-S1cos2ω1(η+τ)E2+S1sin2ω1(η+τ).

27 L17(η)=-ω12ω2cos2ω1(η+τ)sin2ω1(η+τ)-1.

Case-6: Let ω0=0,ω1>0, then28 L18(η)=4ω1eω1(η+τ)e2ω1(η+τ)-4ω1ω2.

Let ω0=0,ω1>0, then29 L19(η)=4ω1eω1(η+τ)1-4ω1ω2e2ω1(η+τ).

Case-7: Let ω0=0,ω1=0andω2>0, then30 L20(η)=1ω2(η+τ).

Let ω0=0,ω1=0andω2>0, then31 L21(η)=iω2(η+τ).

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 F0,Fj     (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 yields32 M′′(η)+c(c-l)a(b-a)M(η)-2a(b-a)(1-α2M3(η)+N-Sa(b-a)M(η)=0,

and33 la+bc+2ac=0,

solve Eq. (33), we get34 c-lb-a=ca,

putting Eq. (34), into Eq. (32), we get35 M′′(η)+c2a2M(η)-2a(b-a)(1-α2M3(η)+N-Sa(b-a)M(η)=0.

Inserting Eq. (7) into Eqs. (4 and 5), and then integrate, we get the following Eqs.36 S=-bM2(1+α)(b-(1+α)a),

37 N=bM2(1-α)(b-(1-α)a).

Inserting Eqs. (36 and 37), into Eq. (32), acquire an ODE38 M′′(η)+c2a2M(η)-2a(b-a)(1-α2M3(η)=0.

Employing balance principle in Eq. (38), we obtain j=1. The precise results shown in Eq. (9) with J=1 is39 M(η)=F1L(η)+F0.

On comparing the similar powers of L(η))j with j=0,1,2,3,.... We create a system of algebraic equations by combining Eq. (39) with Eq. (38), as well as Eq. (10). After evaluation, we obtain the following results presented below.

Family-1:40 F0→0,F1→-ω2-b2ω1+2ibcω1+α2-1c2ω1,a→-icω1.

It has been established that the aforementioned outcomes are satisfactory to Family 1.41 V1,1(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθsechω1bt+1Γ(ς)ςς-icxω1+Ψω1,

If ω0=0,ω1>0 and ω2≠0, we get

42 R1,1(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςsechω1bt+1Γ(ς)ςς-icxω1+Ψω12(1+α)(b-(1+α)a),

43 U1,1(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςsechω1bt+1Γ(ς)ςς-icxω1+Ψω12(1-α)(b-(1-α)a),

44 V1,2(x,t)=-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθcschω1bt+1Γ(ς)ςς-icxω1+Ψω1,

45 R1,2(x,t)=-b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcschω1bt+1Γ(ς)ςς-icxω1+Ψω12(1+α)(b-(1+α)a),

46 U1,2(x,t)=b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcschω1bt+1Γ(ς)ςς-icxω1+Ψω12(1-α)(b-(1-α)a),

If ω0=0,ω1>0 and ω2=4K1K2, we get

47 V1,3(x,t)=-4k1ω2b2-ω1-2ibcω1-α2-1c2eiθ4k12-ω2coshω1bt+1Γ(ς)ςς-icxω1+Ψ×(4k1ω2b2-ω1-2ibcω1-c2eiθ)1+4k12-ω2sinhω1bt+1Γ(ς)ςς-icxω1+Ψ,

48 R1,3(x,t)=-b-4k1ω2b2-ω1-2ibcω1-α2-1c2eiθ4k12-ω2coshω1bt+1Γ(ς)ςς-icxω1+Ψ+4k12-ω2sinhω1bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

49 U1,3(x,t)=b-4k1ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςς4k12-ω2coshω1bt+1Γ(ς)ςς-icxω1+Ψ+4k12-ω2sinhω1bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

If ω0=ω124ω2,ω1<0 and ω2>0, we get

50 V1,4(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθtanh-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω1,

51 R1,4(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςtanh-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1+α)(b-(1+α)a),

52 U1,4(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςtanh-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1-α)(b-(1-α)a),

53 V1,5(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθcoth-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω1,

54 R1,5(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcoth-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1+α)(b-(1+α)a),

55 U1,5(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcoth-ω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1-α)(b-(1-α)a),

56 V1,6(x,t)=--ω1ω2ω2b2c2eiθtanh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1++isech2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1,

57 R1,6(x,t)=-b--ω1ω2ω2b2c2eiθtanh2bt+1Γ(ς)ςς-icxω1+Ψ+isech2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1+α)(b-(1+α)a),

58 U1,6(x,t)=b--ω1ω2ω2b2c2eiθtanh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ+isech2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1-α)(b-(1-α)a),

59 V1,7(x,t)=--ω1ω2ω2b2c2eiθtanh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1++icoth2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1,

60 R1,7(x,t)=-b--ω1ω2ω2b2-α2-1c2eiθicoth-ω1bt+1Γ(ς)ςς+Ψ22+tanh-ω1bt+1Γ(ς)ςς-icxω1+Ψ2222ω12(1+α)(b-(1+α)a),

61 U1,7(x,t)=b--ω1ω2ω2b2-α2-1c2eiθicoth-ω1bt+1Γ(ς)ςς-icxω1+Ψ22+tanh-ω1bt+1Γ(ς)ςς-icxω1+Ψ2222ω12(1-α)(b-(1-α)a),

62 V1,8(x,t)=--ω1ω2ω2b2-ω1c2eiθE12+E22-E1cosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1E2+E1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ,

63 R1,8(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθE12+E22-E1cosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1E2+E1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

64 U1,8(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθE12+E22-E1cosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1E2+E1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

65 V1,9(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθcosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ+i,

66 R1,9(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ+i2(1+α)(b-(1+α)a),

67 U1,9(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcosh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1sinh2-ω1bt+1Γ(ς)ςς-icxω1+Ψ+i2(1-α)(b-(1-α)a),

If ω0=0,ω1<0 and ω2≠0, we get

68 V1,10(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθsec-ω1bt+1Γ(ς)ςς-icxω1+Ψω1,

69 R1,10(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςsec-ω1bt+1Γ(ς)ςς-icxω1+Ψω12(1+α)(b-(1+α)a),

70 U1,10(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςsec-ω1bt+1Γ(ς)ςς-icxω1+Ψω12(1-α)(b-(1-α)a),

71 V1,11(x,t)=--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθcsc-ω1bt+1Γ(ς)ςς-icxω1+Ψω1,

72 R1,11(x,t)=-b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcsc-ω1bt+1Γ(ς)ςς-icxω1+Ψω12(1+α)(b-(1+α)a),

73 U1,11(x,t)=b--ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcsc-ω1bt+1Γ(ς)ςς-icxω1+Ψω12(1-α)(b-(1-α)a),

If ω0=ω124ω2,ω1>0 and ω2>0, we get

74 V1,12(x,t)=-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςtanω1bt+1Γ(ς)ςς-icxω1+Ψ22ω1,

75 R1,12(x,t)=-b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςtanω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1+α)(b-(1+α)a),

76 U1,12(x,t)=b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςtanω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1-α)(b-(1-α)a),

77 V1,13(x,t)=-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcotω1bt+1Γ(ς)ςς-icxω1+Ψ22ω1,

78 R1,13(x,t)=-b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcotω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1+α)(b-(1+α)a),

79 U1,13(x,t)=b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcotω1bt+1Γ(ς)ςς-icxω1+Ψ22ω12(1-α)(b-(1-α)a),

80 V1,14(x,t)=ω1ω2ω2eiθtan2ω1bt+1Γ(ς)ςς-icxω1+Ψ-sec2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1,

81 R1,14(x,t)=-bω1ω2ω2b2α2-1c2eiθtan2ω1bt+1Γ(ς)ςς-icxω1+Ψ-sec2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1+α)(b-(1+α)a),

82 U1,14(x,t)=bω1ω2ω2b2α2-1c2eiθtan2ω1bt+1Γ(ς)ςς-icxω1+Ψ-sec2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1-α)(b-(1-α)a),

83 V1,15(x,t)=-ω1ω2ω2b2-αc2eiθtanω1bt+1Γ(ς)ςς-icxω1+Ψ22-cotω1bt+1Γ(ς)ςς-icxω1+Ψ2222ω1,

84 R1,15(x,t)=-b-ω1ω2ω2b2ω1-α2-1c2eiθtanω1bt+1Γ(ς)ςς-icxω1+Ψ22-cotω1bt+1Γ(ς)ςς-icxω1+Ψ2222ω12(1+α)(b-(1+α)a),

85 U1,15(x,t)=b-ω1ω2ω2b2ω1-α2-1c2eiθtanω1bt+1Γ(ς)ςς-icxω1+Ψ22-cotω1bt+1Γ(ς)ςς-icxω1+Ψ2222ω12(1-α)(b-(1-α)a),

86 V1,16(x,t)=-ω1ω2ω2b2-ω1α2-1c2eiθe12-e22-e1cos2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1e2+e1sin2ω1bt+1Γ(ς)ςς-icxω1+Ψ,

87 R1,16(x,t)=-b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθe12-e22-e1cos2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1e2+e1sin2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

88 U1,16(x,t)=b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθe12-e22-e1cos2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1e2+e1sin2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

89 V1,17(x,t)=-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eiθcot2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω1,

90 R1,17(x,t)=-b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcot2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1+α)(b-(1+α)a),

91 U1,17(x,t)=b-ω1ω2ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςcot2ω1bt+1Γ(ς)ςς-icxω1+Ψ2ω12(1-α)(b-(1-α)a),

If ω0=0 and ω1>0, we get

92 V1,18(x,t)=-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ-4ω1ω2+exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ+icx+lt+1Γ(ς)ςς-4ω1ω2+exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ,

93 R1,18(x,t)=-b-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ+icx+lt+1Γ(ς)ςς-4ω1ω2+exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

94 U1,18(x,t)=b-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ+icx+lt+1Γ(ς)ςς-4ω1ω2+exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

95 V1,19(x,t)=-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ1-4ω1ω2exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ×+icx+lt+1Γ(ς)ςς1-4ω1ω2exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ,

96 R1,19(x,t)=-b-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ+icx+lt+1Γ(ς)ςς1-4ω1ω2exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

97 U1,19(x,t)=b-4ω1ω2b2-ω1-2ibcω1-α2-1c2expω1bt+1Γ(ς)ςς-icxω1+Ψ+icx+lt+1Γ(ς)ςς1-4ω1ω2exp2ω1bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

If ω0=0, ω1=0 and ω2>0, we get

98 V1,20(x,t)=-ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1ω2bt+1Γ(ς)ςς-icxω1+Ψ,

99 R1,20(x,t)=-b-ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1ω2bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

100 U1,20(x,t)=b-ω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1ω2bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a),

101 V1,21(x,t)=-iω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1-ω2bt+1Γ(ς)ςς-icxω1+Ψ,

102 R1,21(x,t)=-b-iω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1-ω2bt+1Γ(ς)ςς-icxω1+Ψ2(1+α)(b-(1+α)a),

103 U1,21(x,t)=b-iω2b2-ω1-2ibcω1-α2-1c2eicx+lt+1Γ(ς)ςςω1-ω2bt+1Γ(ς)ςς-icxω1+Ψ2(1-α)(b-(1-α)a).

Dynamical system

A system of dynamic is applied to describe the temporal dependency of a location of the point within its connecting area34. 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, consider104 M′(η)=P,M′′(η)=P′,

After applying the aforementioned transformation to Eq. (104), we obtained the dynamical system that follows105 P=B1,P′=B2=-c2a2M(η)+2α2a2-(b-a)2M3(η).

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 framework35. The graphical representations of sensitivity analysis under appropriate parameter values and initial conditions are shown in Figs. (14-17).Figure 1 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.4andΨ2=1.23, displays the graphical representation of Eq. (41).

Figure 2 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.14andΨ2=1.23, displays the graphical representation of Eq. (44).

Figure 3 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=1.4andΨ2=1.23, displays the graphical representation of Eq. (47).

Figure 4 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=2.4andΨ2=1.23, displays the graphical representation of Eq. (50).

Figure 5 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.34andΨ2=1.23, displays the graphical representation of Eq. (52).

Figure 6 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.9andΨ2=1.23, displays the graphical representation of Eq. (55).

Figure 7 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=1.94andΨ2=1.23, displays the graphical representation of Eq. (58).

Figure 8 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.67andΨ2=1.23, displays the graphical representation of Eq. (61).

Figure 9 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=1.8andΨ2=1.23, displays the graphical representation of Eq. (64).

Figure 10 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.45andΨ2=1.23, displays the graphical representation of Eq. (67).

Figure 11 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.1andΨ2=1.23, displays the graphical representation of Eq. (70).

Figure 12 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.01andΨ2=1.23, displays the graphical representation of Eq. (95).

Figure 13 The values of parameters are ω1=1.5,ω2=3.2,b=2.03,c=1.32,l=1.23,ς=0.03andΨ2=1.23, displays the graphical representation of Eq. (98).

Figure 14 Graphical representation of Eq. (105) with condition (B1,B2)=(2.2,2.8).

Figure 15 Graphical representation of Eq. (105) with condition (B1,B2)=(2.2,2.4).

Figure 16 Graphical representation of Eq. (105) with condition (B1,B2)=(2.3,2.67).

Figure 17 Graphical representation of Eq. (105) with condition (B1,B2)=(0.2,0.5).

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. 1, 2, 3, 4, 5, 6, 7,8, 9, 10, 11, 12 and 13 and every feasible portrait of sensitivity analysis is explored in Figs. 14, 15, 16 and 17.

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.

Graphical description

Fig. (1) demonstrates bright soliton solutions of Eq. (41). Fig. (2) explores singular soliton solutions of Eq. (44). Fig. (3) illustrates kink-type solutions of Eq. (47). Fig. (4) shows dark solutions of Eq. (52). Fig. (5) represents singular solutions of Eq. (58). Fig. (6), illustrates the combo soliton solutions of Eq. (55). Fig. (7), illustrates the singular solutions of Eq. (58). Fig. (8), illustrates the U-shaped singular dark solutions of Eq. (61). Fig. (9) demonstrates bright singular solutions of Eq. (64). Fig. (10) explores periodic solutions of Eq. (67). Fig. (11), illustrates the combo of periodic solutions of Eq. (70). Fig. (12) shows the exponential solutions of Eq. (95). Fig. (13) represents rational solutions of Eq. (98). Figs. 14, 15, 16 and 17, illustrates the physical depiction of sensitivity analysis.

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.

Author contributions

M.N.Investigation, Methodology, Software and Writing-original draft. F.L.Writing-review and editing, and supervision. Y.A. Formal analysis, Conceptualization and Funding Project. This paper has been read and approved by all authors.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Fahim MRA Kundu PR Islam ME Akbar MA Osman M Wave profile analysis of a couple of (3+ 1)-dimensional nonlinear evolution equations by Sine-Gordon expansion approach J. Ocean Eng. Sci. 2022 7 3 272 279 10.1016/j.joes.2021.08.009
Fahim, M. R. A., Kundu, P. R., Islam, M. E., Akbar, M. A. & Osman, M. Wave profile analysis of a couple of (3+ 1)-dimensional nonlinear evolution equations by Sine-Gordon expansion approach. J. Ocean Eng. Sci. 7(3), 272–279 (2022).10.1016/j.joes.2021.08.009
2. Faridi WA Bakar MA Akgül A El-Rahman MA El-Din SM Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches Alex. Eng. J. 2023 78 483 497 10.1016/j.aej.2023.07.049
Faridi, W. A., Bakar, M. A., Akgül, A., El-Rahman, M. A. & El-Din, S. M. Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023).10.1016/j.aej.2023.07.049
3. Garrappa R Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models Commun. Nonlinear Sci. Numer. Simul. 2016 38 178 191 10.1016/j.cnsns.2016.02.015
Garrappa, R. Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models. Commun. Nonlinear Sci. Numer. Simul. 38, 178–191 (2016).10.1016/j.cnsns.2016.02.015
4. Algahtani OJJ Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model Chaos Solitons Fract. 2016 89 552 559 10.1016/j.chaos.2016.03.026
Algahtani, O. J. J. Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fract. 89, 552–559 (2016).10.1016/j.chaos.2016.03.026
5. Rehman HU Inc M Asjad MI Habib A Munir Q New soliton solutions for the space-time fractional modified third order Korteweg-de Vries equation J. Ocean Eng. Sci. 2022 10.1016/j.joes.2022.05.032
Rehman, H. U., Inc, M., Asjad, M. I., Habib, A. & Munir, Q. New soliton solutions for the space-time fractional modified third order Korteweg-de Vries equation. J. Ocean Eng. Sci.10.1016/j.joes.2022.05.032 (2022).10.1016/j.joes.2022.05.032
6. Rehman SU Bilal M Ahmad J The study of solitary wave solutions to the time conformable Schrödinger system by a powerful computational technique Opt. Quant. Electron. 2022 54 4 228 10.1007/s11082-022-03627-6
Rehman, S. U., Bilal, M. & Ahmad, J. The study of solitary wave solutions to the time conformable Schrödinger system by a powerful computational technique. Opt. Quant. Electron. 54(4), 228 (2022).10.1007/s11082-022-03627-6
7. Iqbal I Rehman HU Mirzazadeh M Hashemi MS Retrieval of optical solitons for nonlinear models with Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity Opt. Quant. Electron. 2023 55 7 588 10.1007/s11082-023-04866-x
Iqbal, I., Rehman, H. U., Mirzazadeh, M. & Hashemi, M. S. Retrieval of optical solitons for nonlinear models with Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity. Opt. Quant. Electron. 55(7), 588 (2023).10.1007/s11082-023-04866-x
8. Kumar S Rani S Study of exact analytical solutions and various wave profiles of a new extended (2+ 1)-dimensional Boussinesq equation using symmetry analysis J. Ocean Eng. Sci. 2022 7 5 475 484 10.1016/j.joes.2021.10.002
Kumar, S. & Rani, S. Study of exact analytical solutions and various wave profiles of a new extended (2+ 1)-dimensional Boussinesq equation using symmetry analysis. J. Ocean Eng. Sci. 7(5), 475–484 (2022).10.1016/j.joes.2021.10.002
9. Khan K Akbar MA Study of explicit travelling wave solutions of nonlinear evolution equations Partial Differ. Equ. Appl. Math. 2023 7 100475 10.1016/j.padiff.2022.100475
Khan, K. & Akbar, M. A. Study of explicit travelling wave solutions of nonlinear evolution equations. Partial Differ. Equ. Appl. Math. 7, 100475 (2023).10.1016/j.padiff.2022.100475
10. Wang S Novel soliton solutions of CNLSEs with Hirota bilinear method J. Opt. 2023 52 3 1602 1607 10.1007/s12596-022-01065-x
Wang, S. Novel soliton solutions of CNLSEs with Hirota bilinear method. J. Opt. 52(3), 1602–1607 (2023).10.1007/s12596-022-01065-x
11. Gu Y Zia SM Isam M Manafian J Hajar A Abotaleb M Bilinear method and semi-inverse variational principle approach to the generalized (2+ 1)-dimensional shallow water wave equation Results Phys. 2023 45 106213 10.1016/j.rinp.2023.106213
Gu, Y. et al. Bilinear method and semi-inverse variational principle approach to the generalized (2+ 1)-dimensional shallow water wave equation. Results Phys. 45, 106213 (2023).10.1016/j.rinp.2023.106213
12. Rehman HU Iqbal I Hashemi MS Mirzazadeh M Eslami M Analysis of cubic-quartic-nonlinear Schrödinger’s equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques Optik 2023 287 171028 10.1016/j.ijleo.2023.171028
Rehman, H. U., Iqbal, I., Hashemi, M. S., Mirzazadeh, M. & Eslami, M. Analysis of cubic-quartic-nonlinear Schrödinger’s equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques. Optik 287, 171028 (2023).10.1016/j.ijleo.2023.171028
13. Sivasundari SAS Jeyabarathi P Rajendran L Theoretical analysis of nonlinear equation in reaction-diffusion system: Hyperbolic function method Eur. J. Math. Stat. 2023 4 1 24 31 10.24018/ejmath.2023.4.1.168
Sivasundari, S. A. S., Jeyabarathi, P. & Rajendran, L. Theoretical analysis of nonlinear equation in reaction-diffusion system: Hyperbolic function method. Eur. J. Math. Stat. 4(1), 24–31 (2023).10.24018/ejmath.2023.4.1.168
14. Raza N Salman F Butt AR Gandarias ML Lie symmetry analysis, soliton solutions and qualitative analysis concerning to the generalized q-deformed sinh-gordon equation Commun. Nonlinear Sci. Numer. Simul. 2023 116 106824 10.1016/j.cnsns.2022.106824
Raza, N., Salman, F., Butt, A. R. & Gandarias, M. L. Lie symmetry analysis, soliton solutions and qualitative analysis concerning to the generalized q-deformed sinh-gordon equation. Commun. Nonlinear Sci. Numer. Simul. 116, 106824 (2023).10.1016/j.cnsns.2022.106824
15. Ali A Ahmad J Javed S Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system Alex. Eng. J. 2023 69 121 133 10.1016/j.aej.2023.01.063
Ali, A., Ahmad, J. & Javed, S. Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system. Alex. Eng. J. 69, 121–133 (2023).10.1016/j.aej.2023.01.063
16. Li AG West AC Preindl M Towards unified machine learning characterization of lithium-ion battery degradation across multiple levels: A critical review Appl. Energy 2022 316 119030 10.1016/j.apenergy.2022.119030
Li, A. G., West, A. C. & Preindl, M. Towards unified machine learning characterization of lithium-ion battery degradation across multiple levels: A critical review. Appl. Energy 316, 119030 (2022).10.1016/j.apenergy.2022.119030
17. Zhang S Ba J-M Sun Y-N Dong L A generalized (g’/g)-expansion method for the nonlinear schrödinger equation with variable coefficients Zeitschrift für Naturforschung A 2009 64 11 691 696 10.1515/zna-2009-1104
Zhang, S., Ba, J.-M., Sun, Y.-N. & Dong, L. A generalized (g’/g)-expansion method for the nonlinear schrödinger equation with variable coefficients. Zeitschrift für Naturforschung A 64(11), 691–696 (2009).10.1515/zna-2009-1104
18. Yokus A Isah MA Dynamical behaviors of different wave structures to the Korteweg-de Vries equation with the Hirota bilinear technique Phys. A 2023 622 128819 10.1016/j.physa.2023.128819
Yokus, A. & Isah, M. A. Dynamical behaviors of different wave structures to the Korteweg-de Vries equation with the Hirota bilinear technique. Phys. A 622, 128819 (2023).10.1016/j.physa.2023.128819
19. Alhojilan Y Ahmed HM Novel analytical solutions of stochastic Ginzburg-landau equation driven by wiener process via the improved modified extended tanh function method Alex. Eng. J. 2023 72 269 274 10.1016/j.aej.2023.04.005
Alhojilan, Y. & Ahmed, H. M. Novel analytical solutions of stochastic Ginzburg-landau equation driven by wiener process via the improved modified extended tanh function method. Alex. Eng. J. 72, 269–274 (2023).10.1016/j.aej.2023.04.005
20. Ali MR Khattab MA Mabrouk S Investigation of travelling wave solutions for the (3+ 1)-dimensional hyperbolic nonlinear Schrödinger equation using riccati equation and f-expansion techniques Opt. Quant. Electron. 2023 55 11 991 10.1007/s11082-023-05236-3
Ali, M. R., Khattab, M. A. & Mabrouk, S. Investigation of travelling wave solutions for the (3+ 1)-dimensional hyperbolic nonlinear Schrödinger equation using riccati equation and f-expansion techniques. Opt. Quant. Electron. 55(11), 991 (2023).10.1007/s11082-023-05236-3
21. Eldidamony H Ahmed HM Zaghrout A Ali Y Arnous AH Mathematical methods for construction new soliton solutions of Radhakrishnan-Kundu Lakshmanan equation Alex. Eng. J. 2022 61 9 7111 7120 10.1016/j.aej.2021.12.053
Eldidamony, H., Ahmed, H. M., Zaghrout, A., Ali, Y. & Arnous, A. H. Mathematical methods for construction new soliton solutions of Radhakrishnan-Kundu Lakshmanan equation. Alex. Eng. J. 61(9), 7111–7120 (2022).10.1016/j.aej.2021.12.053
22. Thota S Shanmugasundaram P On new sixth and seventh order iterative methods for solving non-linear equations using homotopy perturbation technique BMC. Res. Notes 2022 15 1 267 10.1186/s13104-022-06154-5 35907910
Thota, S. & Shanmugasundaram, P. On new sixth and seventh order iterative methods for solving non-linear equations using homotopy perturbation technique. BMC. Res. Notes 15(1), 267 (2022).35907910 10.1186/s13104-022-06154-5
23. Noor MA Mohyud-Din ST Modified variational iteration method for solving fourth-order boundary value problems J. Appl. Math. Comput. 2009 29 81 94 10.1007/s12190-008-0090-z
Noor, M. A. & Mohyud-Din, S. T. Modified variational iteration method for solving fourth-order boundary value problems. J. Appl. Math. Comput. 29, 81–94 (2009).10.1007/s12190-008-0090-z
24. Kundu PR Fahim MRA Islam ME Akbar MA The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis Heliyon 2021 7 3 e06459 10.1016/j.heliyon.2021.e06459 33786391
Kundu, P. R., Fahim, M. R. A., Islam, M. E. & Akbar, M. A. The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis. Heliyon 7(3), e06459 (2021).33786391 10.1016/j.heliyon.2021.e06459
25. Gao W Rezazadeh H Pinar Z Baskonus HM Sarwar S Yel G Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique Opt. Quant. Electron. 2020 52 1 13 10.1007/s11082-019-2162-8
Gao, W. et al. Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Opt. Quant. Electron. 52, 1–13 (2020).10.1007/s11082-019-2162-8
26. Sarwar A Gang T Arshad M Ahmed I Ahmad M Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications Ain Shams Eng. J. 2023 14 2 101839 10.1016/j.asej.2022.101839
Sarwar, A., Gang, T., Arshad, M., Ahmed, I. & Ahmad, M. Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications. Ain Shams Eng. J. 14(2), 101839 (2023).10.1016/j.asej.2022.101839
27. Rehman HU Ullah N Imran M Exact solutions of Kudryashov-Sinelshchikov equation using two analytical techniques Eur. Phys. J. Plus 2021 136 6 647 10.1140/epjp/s13360-021-01589-4
Rehman, H. U., Ullah, N. & Imran, M. Exact solutions of Kudryashov-Sinelshchikov equation using two analytical techniques. Eur. Phys. J. Plus 136(6), 647 (2021).10.1140/epjp/s13360-021-01589-4
28. Wazwaz A-M Multiple soliton solutions for an integrable couplings of the Boussinesq equation Ocean Eng. 2013 73 38 40 10.1016/j.oceaneng.2013.08.004
Wazwaz, A.-M. Multiple soliton solutions for an integrable couplings of the Boussinesq equation. Ocean Eng. 73, 38–40 (2013).10.1016/j.oceaneng.2013.08.004
29. Atangana A Baleanu D Alsaedi A Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal Open Phys. 2016 14 1 145 149 10.1515/phys-2016-0010
Atangana, A., Baleanu, D. & Alsaedi, A. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal. Open Phys. 14(1), 145–149 (2016).10.1515/phys-2016-0010
30. Shakeel M Bibi A AlQahtani SA Alawwad AM Dynamical study of a time fractional nonlinear Schrödinger model in optical fibers Opt. Quant. Electron. 2023 55 11 1010 10.1007/s11082-023-05301-x
Shakeel, M., Bibi, A., AlQahtani, S. A. & Alawwad, A. M. Dynamical study of a time fractional nonlinear Schrödinger model in optical fibers. Opt. Quant. Electron. 55(11), 1010 (2023).10.1007/s11082-023-05301-x
31. Houwe A Abbagari S Doka SY Inc M Bouetou TB Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the heisenberg ferromagnetic spin chain Chaos Solitons Fract. 2021 151 111254 10.1016/j.chaos.2021.111254
Houwe, A., Abbagari, S., Doka, S. Y., Inc, M. & Bouetou, T. B. Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the heisenberg ferromagnetic spin chain. Chaos Solitons Fract. 151, 111254 (2021).10.1016/j.chaos.2021.111254
32. Abbagari S Houwe A Doka SY Bouetou TB Inc M Crepin KT W-shaped profile and multiple optical soliton structure of the coupled nonlinear Schrödinger equation with the four-wave mixing term and modulation instability spectrum Phys. Lett. A 2021 418 127710 10.1016/j.physleta.2021.127710
Abbagari, S. et al. W-shaped profile and multiple optical soliton structure of the coupled nonlinear Schrödinger equation with the four-wave mixing term and modulation instability spectrum. Phys. Lett. A 418, 127710 (2021).10.1016/j.physleta.2021.127710
33. Ibrahim S Ashir AM Sabawi YA Baleanu D Realization of optical solitons from nonlinear Schrödinger equation using modified sardar sub-equation technique Opt. Quant. Electron. 2023 55 7 617 10.1007/s11082-023-04776-y
Ibrahim, S., Ashir, A. M., Sabawi, Y. A. & Baleanu, D. Realization of optical solitons from nonlinear Schrödinger equation using modified sardar sub-equation technique. Opt. Quant. Electron. 55(7), 617 (2023).10.1007/s11082-023-04776-y
34. Ali A Ahmad J Javed S Analysis of chaotic structures, bifurcation and soliton solutions to fractional Boussinesq model Phys. Scr. 2023 98 7 075217 10.1088/1402-4896/acdcee
Ali, A. et al. Analysis of chaotic structures, bifurcation and soliton solutions to fractional Boussinesq model. Phys. Scr. 98(7), 075217 (2023).10.1088/1402-4896/acdcee
35. Ali A Ahmad J Javed S Exploring the dynamic nature of soliton solutions to the fractional coupled nonlinear Schrödinger model with their sensitivity analysis Opt. Quant. Electron. 2023 55 9 810 10.1007/s11082-023-05033-y
Ali, A., Ahmad, J. & Javed, S. Exploring the dynamic nature of soliton solutions to the fractional coupled nonlinear Schrödinger model with their sensitivity analysis. Opt. Quant. Electron. 55(9), 810 (2023).10.1007/s11082-023-05033-y
