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Optical Solitons via the Collective Variable Method for the Schrödinger–Hirota Equation

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Abstract

The collective variable (CV) method is an important technique that deals with both conservative and non-conservative systems by revealing set of equations of motion irrespective of their nonlinearities and dissipative terms. Thus, we employ in this paper the CV method for the complete investigation of the Schrödinger–Hirota equation, which plays a vital role in optical communication and fibers. The standard fourth-order Runge–Kutta technique for integration is further sought for the numerical simulation of the resulting system with an emphasis on the pulse width, amplitude, and frequency. Finally, graphical illustrations are supplied to support the results.

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References

  1. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers in anomalous dispersion. Appl. Phys. Lett. 23(3), 142–144 (1973)

    Article  Google Scholar 

  2. Hasegawa, A.: Dynamics of an ensemble of plane waves in nonlinear dispersive media. Phys. Fluids 18(1), 77–79 (1975)

    Article  Google Scholar 

  3. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 13 (1980)

    Article  Google Scholar 

  4. Agarwal, G.P.: Nonlinear Fiber Optic. Academic Press, London (2001)

    Google Scholar 

  5. Ankiewicz, A., Kedziora, D.J., Chowdury, A., Bandelow, U., Akhmediev, N.: Infinite hierarchy of nonlinear Schrodinger equations and their solutions. Phys. Rev. E 93, 012206 (2016)

    Article  MathSciNet  Google Scholar 

  6. Alqahtani, R.T., Babatin, M.M., Biswas, A.: Bright optical solitons for Lakshmanan Porsezian–Daniel model by semi-inverse variational principle. Optik 154, 109–114 (2018)

    Article  Google Scholar 

  7. Savescu, M., Bhrawy, A.H., Hilal, E.M., Alshaery, A.A., Moraru, L., Biswas, A.: Optical solitons in birefringent fibers with four-wave mixing for parabolic law nonlinearity. Optoelectron. Adv. Mater. Rapid Commun. 9, 10 (2015)

    Google Scholar 

  8. Qingjie, C., Tiande, Z., Djidjeli, K., Price, D.W., Twizell, E.H.: Soliton solution of a class of generalized nonlinear Schrodinger equations. Appl. Math. J. Chin. Univ. 12(4), 389–98 (1997)

    Article  MathSciNet  Google Scholar 

  9. Sedeeg, A.K.H., Nuruddeen, R.I., Gomez-Aguilar, J.F.: Generalized optical soliton solutions to the (3+1) dimensional resonant nonlinear Schrödinger equation with Kerr and parabolic law nonlinearities. Opt. Quantum Electron. 51, 173 (2019)

    Article  Google Scholar 

  10. Al Qarni, A.A., Banaja, M.A., Bakodah, H.O.: Numerical analyses optical solitons in dual core couplers with Kerr law nonlinearity. Appl. Math. 6, 1957–1967 (2015)

    Article  Google Scholar 

  11. Bakodah, H.O., Banaja, M.A., Alqarni, A.A., Alshaery, A.A., Younis, M., Zhou, Q., Biswas, A.: Optical solitons in birefringent fibers with Adomian decomposition method. J. Comput. Theor. Nanosci. 12(10), 5846–5853 (2015)

    Article  Google Scholar 

  12. Banaja, M.A., Al Qarni, A.A., Bakodah, H.O., Zhou, Q., Moshokoa, S.P., Biswas, A.: The investigate of optical solitons in cascaded system by improved adomian decomposition scheme. Optik 130, 1107–1114 (2017)

    Article  Google Scholar 

  13. Bernstein, I., Zerrad, E., Zhou, Q., Biswas, A., Melikechi, N.: Dispersive optical solitons with Schrodinger Hirota equation by traveling wave hypothesis. Optoelectron. Adv. Mater. Rapid Commun. 9(5–6), 792–797 (2015)

    Google Scholar 

  14. Bakodah, H.O., Banaja, M.A., Alshaery, A.A., AlQarni, A.A.: Numerical solution of dispersive optical solitons with Schrödinger–Hirota equation by improved Adomian decomposition method. Math. Probl. Eng., ID 2960912 (2019)

  15. Biswas, A., Kara, A.H., Alqahtani, R.T., Ullah, M.Z., Triki, H., Belic, M.: Conservation laws for optical solitons of Lakshmanan–Porsezian–Daniel model. Proc. Roman. Acad. Ser. A 19, 39–44 (2018)

    MathSciNet  Google Scholar 

  16. Wen-Jun, L., et al.: Soliton interaction in the higher-order nonlinear Schrodinger equation investigated with Hirota’s bilinear method. Phys. Rev. E 77(6), 066605 (2008)

    Article  MathSciNet  Google Scholar 

  17. Sachin, K., Singh, K., Gupta, R.K.: Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G/G)-expansion method. Pramana 79(1), 41–60 (2012)

    Article  Google Scholar 

  18. Xianguo, G., Yanyan, L.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dyn. 69(4), 1621–1630 (2012)

    Article  Google Scholar 

  19. Lakshmanan, M., Porsezian, K., Daniel, M.: Effect of discreteness on the continuum limit of the Heisenberg spin chain. Phys. Lett. A 133(9), 483–488 (1988)

    Article  Google Scholar 

  20. Alqahtani, R.T., Babatin, M.M., Biswas, A.: Bright optical solitons for Lakshmanan–Porsezian–Daniel model by semi-inverse variational principle. Optik 154, 109–114 (2018)

    Article  Google Scholar 

  21. Biswas, A., Ekici, M., Sonmezoglu, A., Triki, H., Majid, F.B., Zhou, Q., Moshokoa, S.P., Mirzazadeh, M., Belic, M.: Optical solitons with Lakshmanan–Porsezian–Daniel model using a couple of integration schemes. Optik 158, 705–711 (2018)

    Article  Google Scholar 

  22. Manafian, J., Foroutan, M., Guzali, A.: Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model. Eur. Phys. J. Plus 132, 1–12 (2017)

    Article  Google Scholar 

  23. Biswas, A., Yildirim, Y., Yasar, E., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method. Optik 160, 24–32 (2018)

    Article  Google Scholar 

  24. Moubissi, A.B., Dinda, P.T., Kofane, T.C.: Note on collective variable theory of nonlinear Schrödinger solitons. J. Phys. A: Math. Gen. 33(12), 2453 (2000)

    Article  Google Scholar 

  25. Moubissi, A.B., Nakkeeran, K., Dinda, P.T., Kofane, T.C.: Non-Lagrangian collective variable approach for optical solitons in fibres. J. Phys. A: Math. Gen. 34(1), 129 (2001)

    Article  MathSciNet  Google Scholar 

  26. Shwetanshumala, S.: Temporal solitons in nonlinear media modeled by modified complex Ginzburg Landau equation under collective variable approach. Int. J. Theor. Phys. 48(4), 1122 (2009)

    Article  MathSciNet  Google Scholar 

  27. Veljkovic, M., Milovic, D., Belic, M., Zhou, Q., Moshokoa, S.P., Biswas, A.: Super-sech soliton dynamics in optical metamaterials using collective variables. Facta Univ. Ser. Electron. Energ. 30(1), 39–48 (2016)

    Article  Google Scholar 

  28. Biyoghe, S., Ekogo, T.B., Moubissi, A.B.: Collective variable analysis of the nonlinear Schrödinger equation for soliton molecules in fibers. J. Nonlinear Opt. Phys. Mater. 26(2), 1750023 (2017)

    Article  Google Scholar 

  29. Olivier, A., Diby, A., Yoboué, P., Kamagaté, A.: Spatio-temporal pulsating dissipative Solitons through collective variable methods. J. Appl. Math. Phys. 4, 1032–1041 (2016)

    Article  Google Scholar 

  30. Green, P.D., Milovic, D.M., Lott, D.A., Biswas, A.: Dynamics of Gaussian optical solitons by collective variables method. Appl. Math. Inf. Sci. Int. J. 2(3), 259–273 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Asma, M., Othman, W.A.M., Wong, B.R., Biswas, A.: Chirped optical Gausson perturbation with quadratic-cubic nonlinearity by collective variables. Opt. Quantum Electron. 51, 200 (2019)

    Article  Google Scholar 

  32. Alam, M.N., Akbar, M.A.: Some new exact traveling wave solutions to the simplified MCH equation and the (1+1)-dimensional combined KdV-mKdV equations. J. Assoc. Arab Univ. Basic Appl. Sci. 17, 6–13 (2015)

    Google Scholar 

  33. Roy, R., Akbar, M.A., Wazwaz, A.M.: Exact wave solutions for the nonlinear time fractional Sharma–Tasso–Olver equation and the fractional Klein–Gordon equation in mathematical physics. Opt. Quantum Electron. 50, 25 (2018)

    Article  Google Scholar 

  34. Khan, K., Akbar, M.A.: Solitary wave solutions of some coupled nonlinear evolution equations. J. Sci. Res. 6(2), 273–284 (2014)

    Article  Google Scholar 

  35. Akbar, M.A., Ali, N.M., Zayed, E.M.E.: The generalized and improved (G’/G)-expansion method combined with the Jacobi elliptic equation. Commun. Theor. Phys. 61(6), 669–676 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics; College of Science, University of Bisha, P.O. Box 551, Bisha, 61922, Saudi Arabia

    A. A. Al Qarni

  2. Department of Mathematics, College of Science, King AbdulAziz University, P.O. Box 80327, Jeddah, Saudi Arabia

    A. A. Al Qarni

  3. Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah, Saudi Arabia

    A. A. Alshaery & H. O. Bakodah

Authors
  1. A. A. Al Qarni
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  2. A. A. Alshaery
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  3. H. O. Bakodah
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Correspondence to A. A. Alshaery.

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Al Qarni, A.A., Alshaery, A.A. & Bakodah, H.O. Optical Solitons via the Collective Variable Method for the Schrödinger–Hirota Equation. Int. J. Appl. Comput. Math 7, 8 (2021). https://doi.org/10.1007/s40819-020-00941-z

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