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Bifurcation and chaotic behaviors to the Sasa–Satsuma and higher-order Sasa–Satsuma equations in fluid dynamics and nonlinear optics

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Abstract

A key objective of the paper is to study the dynamical system for the two types of Sasa–Satsuma equations, namely; Sasa–Satsuma equation and higher-order Sasa–Satsuma equation. A Sasa–Satsuma equation is used to describe the propagation of femtosecond pulses through optical fiber systems. The bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For both models, by using the theory of planar dynamical system the bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For a better understanding of these dynamical behaviors, phase portraits in 2D and 3D figures are dawn. For both equations, the equilibrium points and their effects on the bifurcation behavior are analyzed. Moreover, from the presented results, both models have different dynamical behavior.

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References

  • Abdelrahman, M.A.E., Alharbi, A.: Analytical and numerical investigations of the modified Camassa–Holm equation. Pramana 95, 1–9 (2021)

    Article  Google Scholar 

  • Abdelwahed, H.G., Abdelrahman, M.A.E., Alsarhana, A.F., Mohamed, K.: Investigation of the Ripa model via NHRS scheme with its Wide-Ranging applications. Fractal Fract. 6, 745 (2022)

    Article  Google Scholar 

  • Ablowitz, M.J.: Nonlinear dispersive waves: asymptotic analysis and solitons, vol. 47. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  • Ablowitz, M.J., Ablowitz, M.A., Clarkson, P.A., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering, vol. 149. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  • Alharbi, A.R., Almatrafi, M.B.: Analytical and numerical solutions for the variant Boussinseq equations. J. Taibah Univ. Sci. 14, 454–462 (2020)

    Article  Google Scholar 

  • Alharbi, A.R., Faisal, M.I., Shah, F.A., Waseem, M., Ullah, R., Sherbaz, S.: Higher order numerical approaches for nonlinear equations by decomposition technique. IEEE Access 7, 44329–44337 (2019)

    Article  Google Scholar 

  • Almatrafi, M.B., Alharbi, A.R., Tunç, C.: Constructions of the soliton solutions to the good Boussinesq equation. Adv. Differ. Equ. 2020, 1–14 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability. Eur. Phys. J. Plus 132, 1–11 (2017)

    Article  Google Scholar 

  • Chen, S.: Twisted rogue-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 23202 (2013)

    Article  ADS  Google Scholar 

  • Chen, Y.-X., Xiao, X.: Vector soliton pairs for a coupled nonautonomous NLS model with partially nonlocal coupled nonlinearities under the external potentials. Nonlinear Dyn. 109, 2003–2012 (2022)

    Article  Google Scholar 

  • Feng, D., Lü, J., Li, J., He, T.: Bifurcation studies on travelling wave solutions for nonlinear intensity Klein–Gordon equation. Appl. Math. Comput. 189, 271–284 (2007)

    MathSciNet  MATH  Google Scholar 

  • Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–deVries equation. Phys. Rev. Lett. 19, 1095 (1967)

    Article  MATH  ADS  Google Scholar 

  • Geng, K.-L., Mou, D.-S., Dai, C.-Q.: Nondegenerate solitons of 2-coupled mixed derivative nonlinear Schrödinger equations. Nonlinear Dyn. 111, 603–617 (2023)

    Article  Google Scholar 

  • Ghosh, S., Kundu, A., Nandy, S.: Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation. J. Math. Phys. 40, 1993–2000 (1999)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Hosseini, K., Sadri, K., Salahshour, S., Baleanu, D., Mirzazadeh, M., Mustafa Inc.: The generalized Sasa–Satsuma equation and its optical solitons. Opt. Quantum Electron. 54, 723 (2022)

  • Ismael, H.F., Murad, M.A.S., Bulut, H.: M-lump waves and their interaction with multi-soliton solutions for a generalized Kadomtsev–Petviashvili equation in (3+ 1)-dimensions. Chin. J. Phys. 77, 1357–1364 (2022)

    Article  MathSciNet  Google Scholar 

  • Li, J., Zhang, L.: Bifurcations of traveling wave solutions in generalized Pochhammer–Chree equation. Chaos Solitons Fractals 14, 581–593 (2002)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Mao, J.-J., Tian, S.-F., Yan, X.-J., Zhang, T.-T.: Lump solutions and interaction phenomena of the (3+ 1)-dimensional nonlinear evolution equations. Int. J. Numer. Methods Heat Fluid Flow 29, 3417–3436 (2019)

    Article  Google Scholar 

  • Mao, J.-J., Tian, S.-F., Zhang, T.-T.: Rogue waves, homoclinic breather waves and soliton waves for a (3+ 1)-dimensional non-integrable KdV-type equation. Int. J. Numer. Methods Heat Fluid Flow 29, 763–772 (2019)

    Article  Google Scholar 

  • Mao, J.-J., Tian, S.-F., Zou, L., Zhang, T.-T., Yan, X.-J.: Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+ 1)-dimensional B-type Kadomtsev-Petviashvili equation. Nonlinear Dyn. 95, 3005–3017 (2019)

    Article  MATH  Google Scholar 

  • Mohamed, K., Sahmim, S., Benkhaldoun, F., Abdelrahman, M.A.E.: Some recent finite volume schemes for one and two layers shallow water equations with variable density. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9227

    Article  MathSciNet  Google Scholar 

  • Murad, M.A.S., Hamasalh, F.K., Ismael, H.F.: Various exact optical soliton solutions for time fractional Schrödinger equation with second-order spatiotemporal and group velocity dispersion coefficients. Opt. Quantum Electron. 55, 607 (2023)

    Article  Google Scholar 

  • Nimmo, J.J.C., Yilmaz, H.: Binary Darboux transformation for the Sasa–Satsuma equation. J. Phys. A: Math. Theor. 48, 425202 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Shakir, A.P., Sulaiman, T.A., Ismael, H.F., Shah, N.A., Eldin, S.M.: Multiple fusion solutions and other waves behavior to the Broer–Kaup–Kupershmidt system. Alex. Eng. J. 74, 559–567 (2023)

    Article  Google Scholar 

  • Soto-Crespo, J.M., Devine, N., Hoffmann, N.P., Akhmediev, N.: Rogue waves of the Sasa–Satsuma equation in a chaotic wave field. Phys. Rev. E 90, 32902 (2014)

    Article  ADS  Google Scholar 

  • Sulem, C., Sulem, P.-L.: The nonlinear Schrödinger equation. Appl. Math. Sci. 139 (1999)

  • Tarla, S., Ali, K.K., Yilmazer, R., Osman, M.S.: The dynamic behaviors of the Radhakrishnan–Kundu–Lakshmanan equation by Jacobi elliptic function expansion technique. Opt. Quantum Electron. 54, 1–12 (2022)

    Article  Google Scholar 

  • Triki, H., Mirzazadeh, M., Ahmed, H.M., Samir, I., Hashemi, M.S.: Higher-order Sasa–Satsuma equation: Nucci’s reduction and soliton solutions. Eur. Phys. J. Plus 138, 1–10 (2023)

    Article  Google Scholar 

  • Vakhnenko, V.O., Parkes, E.J., Morrison, A.J.: A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons Fractals 17, 683–692 (2003)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Wang, R.-R., Wang, Y.-Y., Dai, C.-Q.: Influence of higher-order nonlinear effects on optical solitons of the complex Swift–Hohenberg model in the mode-locked fiber laser. Opt. Laser Technol. 152, 108103 (2022)

    Article  Google Scholar 

  • Wazwaz, A.-M., Mehanna, M.: Higher-order Sasa–Satsuma equation: bright and dark optical solitons. Optik 243, 167421 (2021)

    Article  ADS  Google Scholar 

  • Wen, X.-K., Jiang, J.-H., Liu, W., Dai, C.-Q.: Abundant vector soliton prediction and model parameter discovery of the coupled mixed derivative nonlinear Schrödinger equation. Nonlinear Dyn. 111, 13343–13355 (2023)

    Article  Google Scholar 

  • Wu, X.-H., Gao, Y.-T., Yu, X., Ding, C.-C.: N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa–Satsuma equation in fluid dynamics and nonlinear optics. Nonlinear Dyn. 111, 16339–16352 (2023)

    Article  Google Scholar 

  • Xu, T., Li, M., Li, L.: Anti-dark and Mexican-hat solitons in the Sasa–Satsuma equation on the continuous wave background. Europhys. Lett. 109, 30006 (2015)

    Article  ADS  Google Scholar 

  • Xu, T., Wang, D., Li, M., Liang, H.: Soliton and breather solutions of the Sasa–Satsuma equation via the Darboux transformation. Phys. Scr. 89, 75207 (2014)

    Article  Google Scholar 

  • Yang, B., Chen, Y.: High-order soliton matrices for Sasa–Satsuma equation via local Riemann–Hilbert problem. Nonlinear Anal. Real World Appl. 45, 918–941 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Yao, S.W., Akinyemi, L., Mirzazadeh, M., Inc, M., Hosseini, K., Şenol, M.: Dynamics of optical solitons in higher-order Sasa–Satsuma equation. Results Phys. 30, 104825 (2021)

    Article  Google Scholar 

  • Zhang, J.-L., Wang, M.-L., Wang, Y.-M., Fang, Z.-D.: The improved F-expansion method and its applications. Phys. Lett. A 350, 103–109 (2006)

    Article  MATH  ADS  Google Scholar 

  • Zhao, L.-C., Li, S.-C., Ling, L.: Rational W-shaped solitons on a continuous-wave background in the Sasa–Satsuma equation. Phys. Rev. E 89, 23210 (2014)

    Article  ADS  Google Scholar 

  • Zhao, L.-C., Li, S.-C., Ling, L.: W-shaped solitons generated from a weak modulation in the Sasa–Satsuma equation. Phys. Rev. E 93, 32215 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  • Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu, W.: Biswas: perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos Solitons Fractals 160, 112198 (2022)

    Article  MATH  Google Scholar 

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  1. Department of Mathematics, College of Science, University of Zakho, Zakho, Kurdistan Region, Iraq

    Hajar F. Ismael

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Ismael, H.F. Bifurcation and chaotic behaviors to the Sasa–Satsuma and higher-order Sasa–Satsuma equations in fluid dynamics and nonlinear optics. Opt Quant Electron 55, 1271 (2023). https://doi.org/10.1007/s11082-023-05529-7

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