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Analyzing dispersive optical solitons in nonlinear models using an analytical technique and its applications

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Abstract

The article focuses on exploring three distinct equations: the Jimbo-Miwa equation (JME), the generalized shallow water equation (GSWE), and the Hirota-Satsuma-Ito equation (HSIE). By applying the \(\exp (-\Phi (\eta ))\)-expansion method (EEM), we have successfully obtained novel solutions with trigonometric, elliptic, and hyperbolic properties. The main objective of this study is to identify and explore previously undiscovered soliton solutions within nonlinear wave equations, contributing to a deeper comprehension of wave behaviors and facilitating potential applications across diverse scientific and engineering domains. The Jimbo-Miwa equation is relevant to integrable systems and mathematical physics, potentially finding applications in quantum field theory and condensed matter physics. The generalized shallow water equation extends the classical shallow water equations, enabling better modeling of complex fluid dynamics like ocean currents and tsunamis. The Hirota-Satsuma-Ito equation, likely a soliton-based nonlinear equation, holds importance in nonlinear optics, fluid dynamics, and possibly biological studies, contributing to the comprehension of wave-like behaviors in diverse systems. Soliton and solitary wave structures are extracted as distinct solutions. By selecting appropriate values for arbitrary parameters within the accurate range, we create 3D, 2D, and contour plots to visualize the discovered solutions. Modifying model parameters enables the alteration of the solution dynamics generated by the models. The calculations for this research were exclusively performed using the symbolic software Mathematica. The solutions received encompass a variety of types, such as dark, bright, combo dark-bright, singular, cuspons, peakons, periodic solitary wave solutions, single-soliton solutions, double-soliton solutions, N-soliton solutions, and numerous others. These solutions have real-life applications in areas such as predicting coastal hazards, improving optical communications, studying nonlinear dynamics, enhancing material science, and advancing medical imaging techniques. The complexity and nonlinear nature of the system are underscored by these findings, emphasizing the necessity for additional analysis. Moreover, the obtained results offer valuable insights into understanding and modeling comparable physical systems. This research marks a significant advancement by utilizing the the \(\exp (-\Phi (\eta ))\)-expansion method to reveal solitonic solutions for an unsolved model, thereby expanding the existing literature and introducing a novel mathematical technique to address nonlinear physical models. The proposed method is concise, transparent, and reliable, leading to reduced computations and widespread applicability.

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References

  • Abdelrahman, M.A., Alkhidhr, H.A.: A robust and accurate solver for some nonlinear partial differential equations and tow applications. Phys. Scr. 95(6), 065212 (2020)

    CAS  Google Scholar 

  • Abro, K.A., Atangana, A., Gomez-Aguilar, J.F.: An analytic study of bioheat transfer pennes model via modern non-integers differential techniques. Eur. Phys. J. Plus 136, 1–11 (2021)

    Google Scholar 

  • Ahmad, J., Mustafa, Z.: Dynamics of exact solutions of nonlinear resonant Schrödinger equation utilizing conformable derivatives and stability analysis. Eur. Phys. J. D 77(6), 123 (2023)

    CAS  ADS  Google Scholar 

  • Ahmad, J., Mustafa, Z., Rezazadeh, H.: New analytical wave structures for some nonlinear dynamical models via mathematical technique. Univ. Wah J. Sci. Technol. (UWJST) 7(1), 51–75 (2023)

    Google Scholar 

  • Ahmad, J., Mustafa, Z., Zulfiqar, A.: Solitonic solutions of two variants of nonlinear Schrödinger model by using exponential function method. Opt. Quant. Electron. 55(7), 633 (2023)

    Google Scholar 

  • Aji, S., Kumam, P., Awwal, A.M., Yahaya, M.M., Kumam, W.: Two hybrid spectral methods with inertial effect for solving system of nonlinear monotone equations with application in robotics. IEEE Access 9, 30918–30928 (2021)

    Google Scholar 

  • Akbar, M.A., Abdullah, F.A., Islam, M.T., Al Sharif, M.A., Osman, M.: New solutions of the soliton type of shallow water waves and superconductivity models. Res. Phys. 44, 106180 (2023)

    Google Scholar 

  • Ali, A., Ahmad, J., Javed, S., Rehman, S.-U.: Analysis of chaotic structures, bifurcation and soliton solutions to fractional boussinesq model. Physica Scripta (2023)

  • Almutairi, A., El-Metwally, H., Sohaly, M., Elbaz, I.: Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology. Adv. Diff. Eq. 2021, 1–32 (2021)

    MathSciNet  Google Scholar 

  • Alquran, M., Jaradat, I.: A novel scheme for solving caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn. 91, 2389–2395 (2018)

    Google Scholar 

  • Andrade, J. H., Wei, J.: Classification for positive singular solutions to critical sixth order equations. arXiv preprint arXiv:2210.04376, 9 (2022)

  • Andreeva, E. I., Potapov, I. A.: Possibilities of using optical solitons in high-speed systems. In International Youth Conference on Electronics, Telecommunications and Information Technologies: Proceedings of the YETI 2020, St. Petersburg, Russia, pages 241–245. Springer (2020)

  • Attia, R. A., Xia, Y., Zhang, X., Khater, M. M.: Analytical and numerical investigation of soliton wave solutions in the fifth-order kdv equation within the kdv-kp framework. Res. Phys. 106646 (2023)

  • Bainov, D., Simeonov, P.: Impulsive differential equations: periodic solutions and applications. Routledge (2017)

    Google Scholar 

  • Beck, C.E.W., Jentzen, A.: Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. J. Nonlinear Sci. 29, 1563–1619 (2019)

    MathSciNet  ADS  Google Scholar 

  • Cachazo, F., Umbert, B., Zhang, Y.: Singular solutions in soft limits. J. High Energy Phys. 2020(5), 1–33 (2020)

    MathSciNet  Google Scholar 

  • Chen, S.-J., Ma, W.-X., Lü, X.: Bäcklund transformation, exact solutions and interaction behaviour of the (3+ 1)-dimensional hirota-satsuma-ito-like equation. Commun. Nonlinear Sci. Num. Simul. 83, 105135 (2020)

    Google Scholar 

  • Chen, X., Liu, Y., Zhuang, J.: Soliton solutions and their degenerations in the (2+ 1)-dimensional ZHirota-satsuma-ito equations with time-dependent linear phase speed. Nonlinear Dyn. 111(11), 10367–10380 (2023)

    Google Scholar 

  • Deng, G.-F., Gao, Y.-T., Ding, C.-C., Su, J.-J.: Solitons and breather waves for the generalized konopelchenko-dubrovsky-kaup-kupershmidt system in fluid mechanics, ocean dynamics and plasma physics. Chaos Solitons Fractals 140, 110085 (2020)

    MathSciNet  Google Scholar 

  • Duran, S.: Travelling wave solutions and simulation of the lonngren wave equation for tunnel diode. Opt. Quant. Electron. 53(8), 458 (2021)

    Google Scholar 

  • Duran, S., Kaya, D.: Breaking analysis of solitary waves for the shallow water wave system in fluid dynamics. Eur. Phys. J. Plus 136(9), 1–12 (2021)

    Google Scholar 

  • Gilpin, W., Huang, Y., Forger, D.B.: Learning dynamics from large biological data sets: machine learning meets systems biology. Curr. Opin. Syst. Biol. 22, 1–7 (2020)

    Google Scholar 

  • He, J.-H., El-Dib, Y.O.: Homotopy perturbation method with three expansions. J. Math. Chem. 59, 1139–1150 (2021)

    MathSciNet  CAS  Google Scholar 

  • Hu, J.-Y., Feng, X.-B., Yang, Y.-F.: Optical envelope patterns perturbation with full nonlinearity for gerdjikov-ivanov equation by trial equation method. Optik 240, 166877 (2021)

    ADS  Google Scholar 

  • Iqbal, A., Naeem, I.: Generalized compacton equation, conservation laws and exact solutions. Chaos Solitons Fractals 154, 111604 (2022)

    MathSciNet  Google Scholar 

  • Kassem, M., Rashed, A.: N-solitons and cuspon waves solutions of (2+ 1)-dimensional broer-kaup-kupershmidt equations via hidden symmetries of lie optimal system. Chin. J. Phys. 57, 90–104 (2019)

    MathSciNet  Google Scholar 

  • Khalique, C.M., Plaatjie, K.: Exact solutions and conserved vectors of the two-dimensional generalized shallow water wave equation. Mathematics 9(12), 1439 (2021)

    Google Scholar 

  • Khater, M., Ahmed, A.E.-S.: Strong Langmuir turbulence dynamics through the trigonometric quintic and exponential b-spline schemes. AIMS Math. 6(6), 5896–5908 (2021)

    MathSciNet  Google Scholar 

  • Khater, M.M.: Diverse bistable dark novel explicit wave solutions of cubic-quintic nonlinear Helmholtz model. Mod. Phys. Lett. B 35(26), 2150441 (2021)

    MathSciNet  CAS  ADS  Google Scholar 

  • Khater, M.M.: Numerical simulations of Zakharov’s (zk) non-dimensional equation arising in Langmuir and ion-acoustic waves. Mod. Phys. Lett. B 35(31), 2150480 (2021)

    MathSciNet  CAS  ADS  Google Scholar 

  • Khater, M.M., Ahmed, A.E.-S., Alfalqi, S., Alzaidi, J.: Diverse novel computational wave solutions of the time fractional kolmogorov-petrovskii-piskunov and the (2+ 1)-dimensional zoomeron equations. Phys. Scr. 96(7), 075207 (2021)

    ADS  Google Scholar 

  • Khater, M.M., Alabdali, A.M.: Multiple novels and accurate traveling wave and numerical solutions of the (2+ 1) dimensional fisher-kolmogorov-petrovskii-piskunov equation. Mathematics 9(12), 1440 (2021)

    Google Scholar 

  • Khater, M.M., Elagan, S., El-Shorbagy, M., Alfalqi, S., Alzaidi, J., Alshehri, N.A.: Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation. Commun. Theor. Phys. 73(9), 095003 (2021)

    MathSciNet  ADS  Google Scholar 

  • Khater, M.M., Lu, D.: Analytical versus numerical solutions of the nonlinear fractional time space telegraph equation. Mod. Phys. Lett. B 35(19), 2150324 (2021)

    MathSciNet  CAS  ADS  Google Scholar 

  • Khater, M.M., Nofal, T.A., Abu-Zinadah, H., Lotayif, M.S., Lu, D.: Novel computational and accurate numerical solutions of the modified Benjamin-bona-Mahony (bbm) equation arising in the optical illusions field. Alex. Eng. J. 60(1), 1797–1806 (2021)

    Google Scholar 

  • Khodadadi, V., Rahatabad, F.N., Sheikhani, A., Dabanloo, N.J.: Nonlinear analysis of biceps surface EMG signals for chaotic approaches. Chaos Solitons Fractals 166, 112965 (2023)

    MathSciNet  Google Scholar 

  • Kumar, D., Kumar, S.: Some new periodic solitary wave solutions of (3+ 1)-dimensional generalized shallow water wave equation by lie symmetry approach. Comput. Math. Appl. 78(3), 857–877 (2019)

    MathSciNet  Google Scholar 

  • Kumar, S., Jadaun, V., Ma, W.X.: Application of the lie symmetry approach to an extended Jimbo-Miwa equation in (3+ 1) dimensions. Eur. Phys. J. Plus 136, 1–30 (2021)

    Google Scholar 

  • Kumar, S., Kumar, A., Wazwaz, A.-M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135(11), 1–17 (2020)

    Google Scholar 

  • Li, W., Akinyemi, L., Lu, D., Khater, M.M.: Abundant traveling wave and numerical solutions of weakly dispersive long waves model. Symmetry 13(6), 1085 (2021)

    ADS  Google Scholar 

  • Liang, X., Cai, Z., Wang, M., Zhao, X., Chen, H., Li, C.: Chaotic oppositional sine-cosine method for solving global optimization problems. Eng. Comput. 1–17 (2022)

  • Liu, J.-G., Osman, M.: Nonlinear dynamics for different non-autonomous wave structures solutions of a 3d variable-coefficient generalized shallow water wave equation. Chin. J. Phys. 77, 1618–1624 (2022)

    Google Scholar 

  • Long, F., Alsallami, S.A., Rezaei, S., Nonlaopon, K., Khalil, E.: New interaction solutions to the (2+ 1)-dimensional Hirota-satsuma-ito equation. Res. Phys. 37, 105475 (2022)

    Google Scholar 

  • Ma, Y.-L., Li, B.-Q.: Soliton resonances for a transient stimulated Raman scattering system. Nonlinear Dyn. 111(3), 2631–2640 (2023)

    Google Scholar 

  • Malik, S., Hashemi, M.S., Kumar, S., Rezazadeh, H., Mahmoud, W., Osman, M.: Application of new Kudryashov method to various nonlinear partial differential equations. Opt. Quant. Electron. 55(1), 8 (2023)

    Google Scholar 

  • Paliathanasis, A.: Lie symmetries and singularity analysis for generalized shallow-water equations. Int. J. Nonlinear Sci. Num. Simul. 21(7–8), 739–747 (2020)

    MathSciNet  Google Scholar 

  • Rani, A., Ashraf, M., Ahmad, J., Ul-Hassan, Q.M.: Soliton solutions of the Caudrey-Dodd-Gibbon equation using three expansion methods and applications. Opt. Quant. Electron. 54(3), 158 (2022)

    Google Scholar 

  • Rani, A., Zulfiqar, A., Ahmad, J., Hassan, Q.M.U.: New soliton wave structures of fractional Gilson-pickering equation using tanh-coth method and their applications. Res. Phys. 29, 104724 (2021)

    Google Scholar 

  • Rasool, T., Hussain, R., Rezazadeh, H., Gholami, D.: The plethora of exact and explicit soliton solutions of the hyperbolic local (4+ 1)-dimensional blmp model via gerf method. Res. Phys. 46, 106298 (2023)

    Google Scholar 

  • Raza, N., Arshed, S.: Chiral bright and dark soliton solutions of Schrodinger’s equation in (1+ 2)-dimensions. Ain Shams Eng. J. 11(4), 1237–1241 (2020)

    Google Scholar 

  • Schuwirth, N., Borgwardt, F., Domisch, S., Friedrichs, M., Kattwinkel, M., Kneis, D., Kuemmerlen, M., Langhans, S.D., Martínez-López, J., Vermeiren, P.: How to make ecological models useful for environmental management. Ecol. Model. 411, 108784 (2019)

    Google Scholar 

  • Shams, M., Kausar, N., Samaniego, C., Agarwal, P., Ahmed, S. F., Momani, S.: On efficient fractional caputo-type simultaneous scheme for finding all roots of polynomial equations with biomedical engineering applications. Fractals, page 2340075 (2023)

  • Takembo, C.N., Mvogo, A., Ekobena Fouda, H.P., Kofané, T.C.: Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95, 1067–1078 (2019)

    Google Scholar 

  • Tariq, K.U., Ahmed, A., Ma, W.-X.: On some soliton structures to the schamel-korteweg-de vries model via two analytical approaches. Mod. Phys. Lett. B 36(226n27), 2250137 (2022)

    MathSciNet  CAS  ADS  Google Scholar 

  • Tariq, K.U., Wazwaz, A., Kazmi, S.R.: On the dynamics of the (2+ 1)-dimensional chiral nonlinear Schrödinger model in physics. Optik 285, 170943 (2023)

    ADS  Google Scholar 

  • Tariq, K.U., Wazwaz, A., Tufail, R.: Lump, periodic and travelling wave solutions to the (2+ 1)-dimensional pkp-bkp model. Eur. Phys. J. Plus 137(10), 1–22 (2022)

    Google Scholar 

  • Tariq, K.U., Wazwaz, A.-M., Javed, R.: Construction of different wave structures, stability analysis and modulation instability of the coupled nonlinear drinfel’d-sokolov-wilson model. Chaos Solitons Fractals 166, 112903 (2023)

    MathSciNet  Google Scholar 

  • Tariq, K.U.-H., Seadawy, A.R.: Soliton solutions of (3+ 1)-dimensional korteweg-de vries benjamin-bona-mahony, kadomtsev-petviashvili benjamin-bona-mahony and modified korteweg de vries-zakharov-kuznetsov equations and their applications in water waves. J. King Saud Univ. Sci. 31(1), 8–13 (2019)

    Google Scholar 

  • Xu, H.N., Ruan, W.Y., Zhang, Y., Lu, X.: Multi-exponential wave solutions to two extended jimbo-miwa equations and the resonance behavior. Appl. Math. Lett. 99, 105976 (2020)

    MathSciNet  Google Scholar 

  • Yan, L., Yel, G., Kumar, A., Baskonus, H.M., Gao, W.: Newly developed analytical scheme and its applications to the some nonlinear partial differential equations with the conformable derivative. Fractal Fract. 5(4), 238 (2021)

    Google Scholar 

  • Yang, C., Liu, W., Zhou, Q., Mihalache, D., Malomed, B.A.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Google Scholar 

  • Yang, J.Y., Ma, W.X.: Abundant lump-type solutions of the jimbo-miwa equation in (3+ 1)-dimensions. Comput. Math. Appl. 73, 220–225 (2017)

    MathSciNet  Google Scholar 

  • Yang, X., Zhang, Z., Wazwaz, A.-M., Wang, Z.: A direct method for generating rogue wave solutions to the (3+ 1)-dimensional korteweg-de vries benjamin-bona-mahony equation. Phys. Lett. A 449, 128355 (2022)

    MathSciNet  CAS  Google Scholar 

  • Yin, T., Xing, Z., Pang, J.: Modified hirota bilinear method to (3+ 1)-d variable coefficients generalized shallow water wave equation. Nonlinear Dyn. 111(11), 9741–9752 (2023)

    Google Scholar 

  • Yokuş, A., Durur, H., Duran, S., Islam, M.T.: Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism. Comput. Appl. Math. 41(4), 174 (2022)

    MathSciNet  Google Scholar 

  • Yong-Yan, F., Manafian, J., Zia, S.M., Huy, D.T.N., Le, T.-H.: Analytical treatment of the generalized hirota-satsuma-ito equation arising in shallow water wave. Adv. Math. Phys. 2021, 1–26 (2021)

    MathSciNet  Google Scholar 

  • Younis, M., Ali, S., Rizvi, S.T.R., Tantawy, M., Tariq, K.U., Bekir, A.: Investigation of solitons and mixed lump wave solutions with (3+ 1)-dimensional potential-ytsf equation. Commun. Nonlinear Sci. Num. Simul. 94, 105544 (2021)

    MathSciNet  Google Scholar 

  • Zhou, Y., Manukure, S., Ma, W.-X.: Lump and lump-soliton solutions to the hirota-satsuma-ito equation. Commun. Nonlinear Sci. Num. Simul. 68, 56–62 (2019)

    MathSciNet  Google Scholar 

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  1. Department of Mathematics, Faculty of Science, University of Gujrat, Gujrat, 50700, Pakistan

    Jamshad Ahmad, Zulaikha Mustafa & Jamila Habib

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  1. Jamshad Ahmad
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JA: Resources, acquisition, Supervision, Writing—review and editing, Validation. ZM: Conceptualization, Methodology, Software, Writing—original draft. JH: Visualization, Investigation, Writing-review and editing.

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Correspondence to Jamshad Ahmad.

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Ahmad, J., Mustafa, Z. & Habib, J. Analyzing dispersive optical solitons in nonlinear models using an analytical technique and its applications. Opt Quant Electron 56, 77 (2024). https://doi.org/10.1007/s11082-023-05552-8

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