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Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation

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Abstract

The purpose of this manuscript is to investigate the precise soliton solution of the \((1 + 1)\)-dimensional Sharma–Tasso–Olver–Burgers equation. This study thoroughly explores the characteristics and solutions of this nonlinear equation. To pinpoint the exact soliton solution, we employ analytical techniques, specifically the \(\exp (-\phi (\xi ))\)-expansion method, as well as stability and sensitivity analysis. We begin by introducing the equation and discussing its role in mathematical modeling. It’s worth noting that the equation’s inherent nonlinearity adds complexity and poses challenges for analysis and solution. Our research objective is to identify solutions through graphical interpretation, allowing us to gain insights into their behavior. To achieve this, we utilize relevant parameter values to emphasize the physical properties of the provided data. We use the Mathematica and Maple software to demonstrate the solutions that have been discovered in 3D, 2D, and contour plots for the purpose of physical expression and graphical representation. The investigated system is shown to be stable since a little change in the initial conditions does not result in an immense shift in solutions. The sensitivity and stability analyses are also provided at various initial conditions. Importantly, these methods have broader applications and are valuable for conveying nonlinear physical models in the field of nonlinear sciences, not limited to the Sharma–Tasso–Olver–Burgers equation alone.

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References

  • Ahmed, M.S., Zaghrout, A.S., Ahmed, H.M., Arnous, A.H.: Optical soliton perturbation of the Gerdjikov-Ivanov equation with spatio-temporal dispersion using a modified extended direct algebraic method. Optik 259, 168904 (2022)

    ADS  Google Scholar 

  • Akinyemi, L., Mirzazadeh, M., Amin, B.S., Hosseini, K.: Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov’s law of refractive index using the first integral method. J. Mod. Opt. 69(3), 172–182 (2022)

    ADS  MathSciNet  Google Scholar 

  • Arefin, M.A., Khatun, M.A., Uddin, M.H., İnç, M.: Investigation of adequate closed-form travelling wave solution to the space-time fractional non-linear evolution equations. J. Ocean Eng. Sci. 7(3), 292–303 (2022)

    Google Scholar 

  • Arefin, M.A., Khatun, M.A., Islam, M.S., Akbar, M.A., Uddin, M.H.: Explicit soliton solutions to the fractional order nonlinear models through the Atangana Beta derivative. Int. J. Theor. Phys. 62(6), 134 (2023)

    MathSciNet  MATH  Google Scholar 

  • Cinar, M., Secer, A., Ozisik, M., Bayram, M.: Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using sardar sub-equation method. Opt. Quant. Electron. 54(7), 402 (2022)

    Google Scholar 

  • Das, P.K., Mirhosseini, A.S.M., Gholami, D., Rezazadeh, H.: A comparative study between obtained solutions of the coupled Fokas-Lenells equations by Sine-Gordon expansion method and rapidly convergent approximation method. Optik 283, 170888 (2023)

    ADS  Google Scholar 

  • Duran, S., Durur, H.: journal=Mathematical Methods in the Applied Sciences volume=46 number=8 pages=8915–8933 year= publisher=Wiley Online Library Yokuş, A. Traveling wave and general form solutions for the coupled higgs system (2023)

  • Durur, H., Yokuş, A., Duran, S.: Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions. Int. J. Modern Phys. B 2450054 (2023)

  • Ekici, M.: Optical solitons with Kudryashov s quintuple power-law coupled with dual form of non-local law of refractive index with extended Jacobi s elliptic function. Opt. Quant. Electron. 54(5), 279 (2022)

    Google Scholar 

  • Ekici, M.: Stationary optical solitons with Kudryashov s quintuple power law nonlinearity by extended Jacobi s elliptic function expansion. J. Nonlinear Opt. Phys. Mater. 32(01), 2350008 (2023)

    ADS  Google Scholar 

  • Ekici, M., Sonmezoglu, A.: Optical solitons with Biswas-Arshed equation by extended trial function method. Optik 177, 13–20 (2019)

    ADS  Google Scholar 

  • Ghayad, M.S., Badra, N.M., Ahmed, H.M., Rabie, W.B.: Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method. Alex. Eng. J. 64, 801–811 (2023)

    Google Scholar 

  • Hosseini, K., Akbulut, A.R.Z.U., Baleanu, D., Salahshour, S.: The Sharma–Tasso–Olver-burgers equation: its conservation laws and kink solitons. Commun. Theor. Phys. 74(2), 025001 (2022)

    ADS  MathSciNet  MATH  Google Scholar 

  • Houwe, A., Rezazadeh, H., Bekir, A., Doka, S.Y.: Traveling-wave solutions of the Klein–Gordon equations with m-fractional derivative. Pramana 96(1), 26 (2022)

    ADS  Google Scholar 

  • 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)

    Google Scholar 

  • Islam, M.R., et al.: Application of exp (phi (xi))-expansion method for Tzitzeica type nonlinear evolution equations. J. Found. Appl. Phys. 4(1), 8–18 (2016)

    MathSciNet  Google Scholar 

  • Islam, S.M.R., Khan, K., Akbar, M.A.: Study of exp (-[phi]([xi]))-expansion method for solving nonlinear partial differential equations. Br. J. Math. Comput. Sci 5(3), 397 (2015)

    Google Scholar 

  • Ismael, H.F., Sulaiman, T.A.: On the dynamics of the nonautonomous multi-soliton, multi-lump waves and their collision phenomena to a (3+ 1)-dimensional nonlinear model. Chaos Solitons Fractals 169, 113213 (2023)

    MathSciNet  Google Scholar 

  • Kemaloğlu, B., Yel, G., Bulut, H.: An application of the rational sine-Gordon method to the Hirota equation. Opt. Quant. Electron. 55(7), 658 (2023)

    Google Scholar 

  • Khalil, T.A., Badra, N., Ahmed, H.M., Rabie, W.B.: Optical solitons and other solutions for coupled system of nonlinear Biswas-Milovic equation with Kudryashov s law of refractive index by Jacobi elliptic function expansion method. Optik 253, 168540 (2022)

    ADS  Google Scholar 

  • Khan, K., Akbar, M.A.: The exp (- \(\phi\) (\(\xi\)))-expansion method for finding travelling wave solutions of Vakhnenko–Parkes equation. Int. J. Dyn. Syst. Differ. Equ. 5(1), 72–83 (2014)

    MathSciNet  MATH  Google Scholar 

  • Khater, M.M.A.: Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the exp (- \(\varphi\) (\(\xi\)))-expansion method. Cogent Math. 3(1), 1172397 (2016)

    MathSciNet  MATH  Google Scholar 

  • Khater, M.M.A.: Nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities; computational simulations. Chaos Solitons Fractals 157, 111970 (2022)

    MathSciNet  MATH  Google Scholar 

  • Khatun, M.A., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Numerous explicit soliton solutions to the fractional simplified Camassa-Holm equation through two reliable techniques. Ain Shams Eng. J. 102214 (2023)

  • Li, J.: Geometric properties and exact traveling wave solutions for the generalized Burger-Fisher equation and the Sharma–Tasso–Olver equation. J. Nonlinear Model. Anal. 1(1), 1–10 (2019)

    Google Scholar 

  • Liu, H.: An equivalent form for the\(\backslash\)exp (-\(\phi\) (\(\xi\))) exp (-\(\phi\) (\(\xi\)))-expansion method. Jpn. J. Ind. Appl. Math. 35, 1153–1161 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  • Lü, X., Chen, S.J.: Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dyn. 103, 947–977 (2021)

    MATH  Google Scholar 

  • Lu, D., Seadawy, A.R., Wang, J., Arshad, M., Farooq, U.: Soliton solutions of the generalised third-order nonlinear Schrödinger equation by two mathematical methods and their stability. Pramana 93, 1–9 (2019)

    Google Scholar 

  • Mahmood, I., Hussain, E., Mahmood, A., Anjum, A., Shah, S.A.A.: Optical soliton propagation in the Benjamin–Bona–Mahoney–Peregrine equation using two analytical schemes. Optik 171099 (2023)

  • Miao, Z., Hu, X., Chen, Y.: Interaction phenomenon to (1+ 1)-dimensional Sharma–Tasso–Olver–Burgers equation. Appl. Math. Lett. 112, 106722 (2021)

    MathSciNet  MATH  Google Scholar 

  • Rao, J., Mihalache, D., Cheng, Y., He, J.: Lump-soliton solutions to the Fokas system. Phys. Lett. A 383(11), 1138–1142 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  • Rezazadeh, H., Davodi, A.G., Gholami, D.: Combined formal periodic wave-like and soliton-like solutions of the conformable Schrödinger-KdV equation using the G’/G-expansion technique. Results Phys. 47, 106352 (2023)

    Google Scholar 

  • Sadiya, U., Inc, M., Arefin, M.A., Uddin, M.H.: Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach. J. Taibah Univ. Sci. 16(1), 594–607 (2022)

    Google Scholar 

  • Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: N-soliton, m th-order breather, hth-order lump, and hybrid solutions of an extended \((3+ 1)\)-dimensional Kadomtsev–Petviashvili equation. Nonlinear Dyn. pp 1–18 (2023)

  • Sonmezoglu, A.: Stationary optical solitons having Kudryashovs quintuple power law nonlinearity by extended \(\frac{G^{\prime }}{G}\)-expansion. Optik 253, 168521 (2022)

    ADS  Google Scholar 

  • Tiecheng, X., Xiaohong, C., Dengyuan, C.: Darboux transformation and soliton-like solutions of nonlinear Schroedinger equations. Chaos Solitons Fractals 26, 889–896 (2005)

    MathSciNet  MATH  Google Scholar 

  • Uddin, M.H., Arefin, M.A., Akbar, M.A., Inc, M.: New explicit solutions to the fractional-order burgers’ equation. Math. Probl. Eng. 2021, 1–11 (2021)

    MathSciNet  MATH  Google Scholar 

  • Verma, P., Kaur, L.: New exact solutions of the (\(4+1\))-dimensional Fokas equation via extended version of exp (\(-\psi\) (\(\kappa\)))-expansion method. Int. J. Appl. Comput. Math. 7(3), 104 (2021)

    MATH  Google Scholar 

  • Wang, M.Y., Biswas, A., Yıldırım, Y., Moraru, L., Moldovanu, S., Alshehri, H.M.: Optical solitons for a concatenation model by Trial equation approach. Electronics 12(1), 19 (2023)

    Google Scholar 

  • Yan, Z., Lou, S.: Soliton molecules in Sharma–Tasso–Olver–Burgers equation. Appl. Math. Lett. 104, 106271 (2020)

    MathSciNet  MATH  Google Scholar 

  • Yao, S.W., Behera, S., Inc, M., Rezazadeh, H., Virdi, J.P.S., Mahmoud, W.H., Arqub, O.A., Osman, M.S.: Analytical solutions of conformable Drinfel d-Sokolov-Wilson and Boiti Leon pempinelli equations via sine-cosine method. Results Phys. 42, 105990 (2022)

    Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Yokus, A., Iskenderoglu, G., Kaya, D.: Application of some nonclassical methods for p-defocusing complex Klein-Gordon equation. Opt. Quant. Electron. 55(5), 403 (2023)

    Google Scholar 

  • Yu, J., Ren, B., Wang, W.: Symmetry reductions, cte method and interaction solutions for Sharma–Tasso–Olver–Burgers equation. Symmetry 14(8), 1690 (2022)

    ADS  Google Scholar 

  • Zaman, U.H.M., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique. PLoS ONE 18(5), e0285178 (2023)

    Google Scholar 

  • Zaman, U.H.M., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Stable and effective traveling wave solutions to the non-linear fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony equations. Partial Differ. Equ. Appl. Math. 7, 100509 (2023)

    Google Scholar 

  • Zayed, E.M.E.: A note on the modified simple equation method applied to Sharma–Tasso–Olver equation. Appl. Math. Comput. 218(7), 3962–3964 (2011)

    MathSciNet  MATH  Google Scholar 

  • Zhou, Y., Zhuang, J.: Dynamics and exact traveling wave solutions of the Sharma–Tasso–Olver–Burgers equation. Symmetry 14(7), 1468 (2022)

    ADS  Google Scholar 

  • Zhou, T., Tian, B., Chen, Y.: Elastic two-kink, breather, multiple periodic, hybrid and half-/local-periodic kink solutions of a Sharma–Tasso–Olver–Burgers equation for the nonlinear dispersive waves. Qualit. Theory Dyn. Syst. 22(1), 34 (2023)

    MathSciNet  MATH  Google Scholar 

  • Zhujun, J., Jianping, Y., Wei, F.: Bifurcation and chaos in neural excitable system. Chaos Solitons Fractals 27, 197–215 (2006)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the large group Research Project under Grant Number RGP2/37/44.

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Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, 54590, Pakistan

    Ejaz Hussain

  2. College of Computer Science, Chengdu University, Chengdu, 610106, People’s Republic of China

    Zhao Li

  3. Department of Mathematics and Statistics, The University of Lahore, 1-km Defence Road, Lahore, 54000, Pakistan

    Syed Asif Ali Shah

  4. Department of Mathematics, Mutah University, Mutah, Al Karak, Jordan

    Emad A. Az-Zo’bi

  5. Department of Chemistry, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, 61413, Saudi Arabia

    Mohamed Hussien

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  1. Ejaz Hussain
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  2. Zhao Li
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  4. Emad A. Az-Zo’bi
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All authors contributed equally in the preparation, drafting, editing, and reviewing of the manuscript.

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Correspondence to Syed Asif Ali Shah.

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Hussain, E., Li, Z., Shah, S.A.A. et al. Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation. Opt Quant Electron 55, 1274 (2023). https://doi.org/10.1007/s11082-023-05588-w

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