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Controlling optical soliton solutions for higher order Boussinesq equation using bilinear form

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Abstract

The study of solitons including their stability, presence, validity and strength with constant increasing research is not forgettable in this era. In this research paper, we explore solitary wave solutions and dromions for Boussineq equation with the help of Hirota bilinear method (HBM) and Sine-Gordon expansion (SGE) technique. We will enlist these schemes for the study of different types of solitons by implementing HBM and SGE. We will acquire anti dark, multi-solitons, V-shaped, M-shaped, S-shaped and dark travelling wave solutions with their effects during transformations and collision.

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References

  • Ahmad, S., Seadawy, A.R., Rizvi, S.T.R.: Study of breathers, rogue waves and lump solutions for the nonlinear chains of atoms. Opt. Quant. Electron. 54, 320 (2022)

    Google Scholar 

  • Akbar, M.A., Ali, NH Mohd.: Solitary wave solutions of fourth order Boussinesq equation through the \(exp(\phi (\eta ))\)-expansion method. Springer plus 3, 344 (2014)

    Google Scholar 

  • Ali, K., Seadawy, A.R., Ahmad, S., Rizvi, S.T.R.: Discussion on rational solutions for Nematicons in liquid crystal with Kerr law. Chaos, Solitons and Fractals 160, 112218 (2022)

    MathSciNet  MATH  Google Scholar 

  • Alsallami, Shami A.M.., Rizvi, Syed T.R.., Seadawy, Aly R.: Study of Stochastic-Fractional Drinfel’d–Sokolov–Wilson Equation for M-shaped rational, homoclinic breather, periodic and kink-cross rational solutions. Mathematics 11, 1504 (2023)

    Google Scholar 

  • Aziz, N., Seadawy, A.R., Raza, U., Ali, K., Rizvi, S.T.R.: Chirped optical pulses for generalized longitudinal Luglato Lefever: cubic nonlinear Schrödinger equation. Opt. Quant. Electron. 54, 649 (2022)

    Google Scholar 

  • Aziz, N., Seadawy, A.R., Ali, K., Sohail, M., Rizvi, S.T.R.: The nonlinear Schrödinger equation with polynomial law nonlinearity: localized chirped optical and solitary wave solutions. Opt. Quant. Electron. 54, 458 (2022)

    Google Scholar 

  • Bashir, A., Seadawy, A.R., Ahmed, S., Rizvi, S.T.R.: The Weierstrass and Jacobi elliptic solutions along with multiwave, homoclinic breather, kink-periodic-cross rational and other solitary wave solutions to Fornberg Whitham equation. Chaos, Solitons and Fractals 163, 112538 (2022)

    MathSciNet  MATH  Google Scholar 

  • Bhrawy, A.H.: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30–46 (2014)

    MathSciNet  MATH  Google Scholar 

  • Bilal, M., Seadawy, Aly R., Younis, M., Rizvi, S.T.R., El-Rashidy, K., Mahmoud, Samy F.: Analytical wave structures in plasma physics modelled by Gilson–Pickering equation by two integration norms. Results Phys. 23, 103959 (2021)

    Google Scholar 

  • Cheemaa, Nadia, Seadawy, Aly R., Chen, Sheng: Some new families of solitary wave solutions of generalized Schamel equation and their applications in plasma physics. Eur. Phys. J. Plus 134(117), 1–10 (2019)

    Google Scholar 

  • Dosser, H.V., Sutherland, B.R.: Weakly nonlinear non Boussinesq internal gravity wavepackets. Physica D 240, 346–356 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ekici, WYu.M., Mirzazadeh, M., Zhou, Q., Liu, W.: Periodic oscillation of dark solitons in nonlinear optics. Optik 165, 341–344 (2018)

    ADS  Google Scholar 

  • Farah, N., Seadawy, A.R., Younis, M., Rizvi, ST Raza.: Interaction properties of soliton molecules and painleve analysis for nano bioelectronics transmission model. Opt. Quant. Electron. 52, 329 (2020)

    Google Scholar 

  • Ghaffar, A., Ali, A., Ahmed, S., Akram, S., Baleanu, D., Nisar, K.S.: A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order. Adv. Diff. Equa. 1, 1–15 (2020)

    MathSciNet  MATH  Google Scholar 

  • Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.J., Stamper-Kurn, D.M., Ketterle, W.: Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392, 151 (1998)

    ADS  Google Scholar 

  • Jabbari, A., Kheiri, H., Bekir, A.: Exact solutions of the coupled Higgs equation and the Maccari system using He’s semi-inverse method and (G’/G)-expansion method. Comput. Math. Appl. 62, 2177–2186 (2011)

    MathSciNet  MATH  Google Scholar 

  • Liu, S., Zhou, Q., Biswas, A., Liu, W.: Phase shift controlling of three solitons in dispersion decreasing fibers. Nonlinear Dyn. 98, 395–401 (2019)

    MATH  Google Scholar 

  • Li, W., Ma, G., Yu, W., Zhang, Y., Liu, M., Yang, C., Liu, W.: Soliton structures in the (1+1)-dimensional Ginzburg-Landau equation with a parity-time-symmetric potential in ultrafast optics. Chin. Phys. B 27(3), 030504 (2018)

    ADS  Google Scholar 

  • Nur, A.M., Ali, AMd., Tauseef, M.D.S.: A novel \((\frac{G}{G^{\prime }})\)-expansion method and its applications to the Boussinesq equation. Chin. Phys. B 23, 2 (2013)

    Google Scholar 

  • Ozkan, Yesim Glam: Emrullah Yaşar and Aly Seadawy, On the multi-waves, interaction and Peregrine-like rational solutions of perturbed Radhakrishnan-Kundu-Lakshmanan equation. Physica Scripta 95(8), 085205 (2020)

    ADS  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Ali, I., Bibi, I., Younis, M.: Chirp-free optical dromions for the presence of higher order spatio-temporal dispersions and absence of self-phase modulation in birefringent fibers. Mod. Phy. Lett. B 34(35), 2050399 (2020)

    ADS  MathSciNet  Google Scholar 

  • Rizvi, S.T.R., Khan, SU Din., Hasssan, M., Fatima, I., Khan, SU Din.: Stable propagation of optical solitons for nonlinear Schrödinger equation with dispersion and self phase modulation. Math. Comput. Simul. 179, 126–136 (2021)

    MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Raza, U.: Chirped optical wave solutions for a nonlinear model with parabolic law and competing weakly nonlocal. Opt. Quant. Electron. 54, 756 (2022)

    Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Batool, T., Ashraf, M.A.: Homoclinic breaters, mulitwave, periodic cross-kink and periodic cross-rational solutions for improved perturbed nonlinear Schrödinger with quadratic-cubic nonlinearity. Chaos Solitons and Fractals 161, 112353 (2022)

    MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Raza, U.: Detailed analysis for chirped pulses to cubic-quintic nonlinear non-paraxial pulse propagation model. J. Geom. Phys. 178, 104561 (2022)

    MathSciNet  MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Farrah, N., Ahmad, S.: Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation. Chaos, Solitons and Fractals 159, 112128 (2022)

    MATH  Google Scholar 

  • Rizvi, S.T.R., Seadawy, A.R., Raza, U.: Some advanced chirped pulses for generalized mixed for nonlinear Schrödinger dynamical equation. Chaos, Solitons and Fractals 163, 112575 (2022)

    MATH  Google Scholar 

  • seadawy, Aly, Kumar, Dipankar, Hosseini, Kamyar, Samadani, F.: The system of equations for the ion sound and Langmuir waves and its new exact solutions. Results Phys. 9, 1631–1634 (2018)

    ADS  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D.: Application of mathematical methods on the ion sound and Langmuir waves dynamical systems. Pramana J. Phys. (2019). https://doi.org/10.1007/s12043-019-1771-x

    Article  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Younis, M., Baleanue, D.: Lump, lump one stripe, multiwaves and breather solutions for the Hunter Sexton equation, Open. Physics 19, 1–20 (2021)

    Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmed, S., Ahmad, A.: Study of dissipative NLSE for dark and bright, multiwave, breather and M-shaped solitons along with some interactions in monochromatic waves. Opt. Quant. Electron. 54, 782 (2022)

    Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Khaliq, A.: Pure-cubic nonlinear Schrödinger model with optical multi peak, homoclinic breathers, periodic-cross-kink and M-shaped solitons. Opt. Quant. Electron. 54, 771 (2022)

    Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmed, Sarfaraz, Bashir, Azhar: Lump solutions, Kuznetsov-Ma breathers, rogue waves and interaction solutions for magneto electro-elastic circular rod. Chaos, Solitons and Fractals 163, 112563 (2022)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Akram, U., Naqvi, S.K.: Optical and analytical solitons to higher order non-Kerr nonlinear Schrödinger dynamical model. J. Geom. Phys. 179, 104616 (2022)

    MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmad, S.: Weierstrass and Jacobi elliptic, bell and kink type, lumps. Ma and Kuznetsov breathers with rogue wave solutions to the dissipative nonlinear Schrödinger equation. Chaos Solitons and Fractals 160, 112258 (2022)

    MATH  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T.R., Ali, K.: Lumps, breathers, interactions and rogue wave solutions for a stochastic gene evolution in double chain deoxyribonucleic acid system. Chaos Solitons and Fractals 161, 112307 (2022)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T.R.: Multiple lump, generalized breathers, Akhmediev breather, manifold periodic and rogue wave solutions for generalized Fitzhugh-Nagumo equation: Applications in nuclear reactor theory. Chaos Solitons and Fractals 161, 112326 (2022)

    MathSciNet  MATH  Google Scholar 

  • Swaters, G.E., Dosser, H.V., Sutherland, B.R.: Conservation laws, Hamiltonian structure, modulational instability properties and solitary wave solutions for a higher order model describing nonlinear internal waves. Stud. Appl. Math. 0, 1–24 (2011)

    MATH  Google Scholar 

  • Tariq, K.U., Seadawy, A.R., Younis, M., Rizvi, S.T.R.: Dispersive traveling wave solutions to the space-time fractional equal-width dynamical equation and its applications. Opt. Quant. Elec. 50, 147 (2018)

    Google Scholar 

  • Wang, C., Nie, Z., Xie, W., Guo, J., Zhou, Q., Liu, W.: Dark soliton control based ondispersion and nonlinearity for third order nonlinear Schrödinger equation. Optik 184, 370–376 (2019)

    ADS  Google Scholar 

  • Yang, Xiangyu, Zhang, Zhao, Wang, Zhen: Degenerate lump wave solutions of the Mel’nikov equation. Nonlinear Dyn. 111, 1553–1563 (2023)

    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 coefficients nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Google Scholar 

  • Yang, Xiangyu, Zhang, Zhao, Li, Wentao, Li, Biao: Breathers, lumps and hybrid solutions of the (2+1)-dimensional Hirota-Satsuma-Ito equation. Rocky Mt. J. Math. 50(1), 319–335 (2020)

    MathSciNet  MATH  Google Scholar 

  • Yang, Xiangyu, Zhang, Zhao, Wazwaz, Abdul-Majid., Wang, Zhen: 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  MATH  Google Scholar 

  • You, Y., Moe, G., Zhang, X., Liu, W.: Interactions between M-typed dark solitons in nonlinear optics. Optik 198, 163–170 (2019)

    Google Scholar 

  • Yu, W., Yang, C., Liu, M., Zhang, Y., Liu, W.: Interactions of solitons, dromions like structures and butterfly-shaped pulses for variable coefficient nonlinear Schrödinger equation. Optik 159, 21–30 (2018)

    ADS  Google Scholar 

  • Zhang, X., Guo, H., Ma, G., Fang, W., Liu, W.: Anti-dark soliton interactions for the variable coefficient nonlinear Schrödinger equation. Optik 161, 217–220 (2018)

    ADS  Google Scholar 

  • Zhang, Y., Yang, C., Yu, W., Liu, M., Ma, G., Liu, W.: Some types of dark solitons interactions in inhomogenous optical fibers. Opt. Quant. Electron. 50, 295 (2018)

    Google Scholar 

  • Zuo, D.W., Zhang, G.F.: Exact solutions of the nonlocal Hirota equations. Appl. Math. Lett. 93, 66–71 (2019)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan

    Syed T. R. Rizvi, Nighat Farah & Sarfraz Ahmad

  2. Mathematics Department, Faculty of Science, Taibah University, Al Madinah Al Munawwarah, 41411, Kingdom of Saudi Arabia

    Aly R. Seadawy

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  1. Syed T. R. Rizvi
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  2. Aly R. Seadawy
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  3. Nighat Farah
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  4. Sarfraz Ahmad
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Correspondence to Aly R. Seadawy.

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Rizvi, S.T.R., Seadawy, A.R., Farah, N. et al. Controlling optical soliton solutions for higher order Boussinesq equation using bilinear form. Opt Quant Electron 55, 865 (2023). https://doi.org/10.1007/s11082-023-04997-1

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