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Dispersive optical soliton solutions of the \((2 + 1)\)-dimensional cascaded system governing by coupled nonlinear Schrödinger equation with Kerr law nonlinearity in plasma

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Abstract

In this article, the Kudryashov \(R\) function approach is employed to derive solitary wave solutions of a \((2 + 1)\)-dimensional cascaded system governed by coupled nonlinear Schrödinger equation with Kerr law nonlinearity using the travelling wave transformation. The method has various features that make symbolic computation much easier, especially when dealing with highly dispersive nonlinear equations. The approach has the advantage of not requiring the usage of a specific function form in computations. The method is especially useful for obtaining exact soliton solutions to nonlinear differential equations of higher order that explain pulse propagation in an optical fibre. The method provides an efficient, easy, and simple procedure for finding solitary wave solutions. Bright and singular soliton solutions have been extracted from the governing equation using this approach, which may be beneficial in various optoelectronic devices such as solar cells, blue lasers, optical couplers, LED traffic lights, photodiodes and magneto-optic waveguides. In order to illustrate the physical significance of the acquired solutions and by giving particular values to undefined parameters, the physical meaning of 3D and 2D geometrical structures for certain derived solutions has been shown here.

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References

  • Ain, Q.T., He, J.H., Anjum, N., Ali, M.: The fractional complex transform: a novel approach to the time-fractional Schrödinger equation. Fractals 28(07), 2050141 (2020)

    Article  ADS  MATH  Google Scholar 

  • Akram, G., Sadaf, M., Dawood, M.: Abundant soliton solutions for Radhakrishnan–Kundu–Laksmanan equation with Kerr law non-linearity by improved -expansion technique. Optik 247, 167787 (2021). https://doi.org/10.1016/j.ijleo.2021.167787

    Article  ADS  Google Scholar 

  • Akram, G., Sadaf, M., Zainab, I.: The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Optik (2022). https://doi.org/10.1016/j.ijleo.2022.168614

    Article  Google Scholar 

  • 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(1), 1–11 (2021)

    Article  MATH  Google Scholar 

  • Albert, D.Z.: Quantum Mechanics and Experience. Harvard University Press (1994)

    Book  Google Scholar 

  • Arcak, M., Angeli, D., Sontag, E.: A unifying integral ISS framework for stability of nonlinear cascades. SIAM J. Control. Optim. 40(6), 1888–1904 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Awan, A.U., Tahir, M., Rehman, H.U.: Singular and bright-singular combo optical solitons in birefringent fibers to the Biswas–Arshed equation. Optik 210, 164489 (2020)

    Article  ADS  Google Scholar 

  • Awan, A.U., Rehman, H.U., Tahir, M., Ramzan, M.: Optical soliton solutions for resonant Schrödinger equation with anti-cubic nonlinearity. Optik 227, 165496 (2021a)

    Article  ADS  Google Scholar 

  • Awan, A.U., Tahir, M., Abro, K.A.: Multiple soliton solutions with chiral nonlinear Schrödinger’s equation in -dimensions. Eur. J. Mech. B Fluids 85, 68–75 (2021b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Biswas, A., Mirzazadeh, M., Triki, H., Zhou, Q., Ullah, M.Z., Moshokoa, S.P., Belic, M.: Perturbed resonant 1-soliton solution with anti-cubic nonlinearity by Riccati-Bernoulli sub-ODE method. Optik 156, 346–350 (2018)

    Article  ADS  Google Scholar 

  • Chabchoub, A., Hoffmann, N., Onorato, M., Slunyaev, A., Sergeeva, A., Pelinovsky, E., Akhmediev, N.: Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86(5), 056601 (2012)

    Article  ADS  Google Scholar 

  • Dahiya, S., Kumar, H., Kumar, A., Gautam, M.S.: New optical dromion and domain wall solutions of cascaded system in -dimensions via various analytical architectures. Int. J. Appl. Comput. Math. 8(3), 1–39 (2022)

    MathSciNet  MATH  Google Scholar 

  • Dan, J., Sain, S., Ghose-Choudhury, A., Garai, S.: Solitary wave solutions of nonlinear PDEs using Kudryashov’s R function method. J. Mod. Opt. 67(19), 1499–1507 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Das, N., Saha Ray, S.: Novel optical soliton solutions for time-fractional resonant nonlinear Schrödinger equation in optical fiber. Opt. Quant. Electron. 54(2), 1–23 (2022)

    Article  Google Scholar 

  • Fibich, G.: The Nonlinear Schrödinger Equation. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  • Ghanbari, B.: Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative. Math. Methods Appl. Sci. 44(11), 8759–8774 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation. Eur. Phys. J. plus 133(4), 1–18 (2018)

    Article  Google Scholar 

  • Guo, B., Ling, L., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85(2), 026607 (2012)

    Article  ADS  Google Scholar 

  • Hashemi, M.S., Akgül, A.: Solitary wave solutions of time-space nonlinear fractional Schrödinger’s equation: two analytical approaches. J. Comput. Appl. Math. 339, 147–160 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • KGonzalez-Gaxiola, O.: Optical soliton solutions for Triki–Biswas equation by Kudryashov’s R function method. Optik 249, 168230 (2022)

    Article  ADS  Google Scholar 

  • Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kudryashov, N.A.: Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik 206, 163550 (2020). https://doi.org/10.1016/j.ijleo.2019.163550

    Article  ADS  Google Scholar 

  • Liaqat, M.I., Akgül, A.: A novel approach for solving linear and nonlinear time-fractional Schrödinger equations. Chaos Solitons Fractals 162, 112487 (2022)

    Article  MATH  Google Scholar 

  • Ling, L., Zhao, L.C.: Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 72, 449–471 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Lou, Y., Zhang, Y., Ye, R., Li, M.: Solitons and dynamics for the integrable nonlocal pair-transition-coupled nonlinear Schrödinger equation. Appl. Math. Comput. 409, 126417 (2021)

    MATH  Google Scholar 

  • Mathanaranjan, T.: Soliton solutions of deformed nonlinear Schrödinger equations using ansatz method. Int. J. Appl. Comput. Math. 7(4), 1–11 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Ortega, R.: Passivity properties for stabilization of cascaded nonlinear systems. Automatica 27(2), 423–424 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Rezazadeh, H., Kurt, A., Tozar, A., Tasbozan, O., Mirhosseini-Alizamini, S.M.: Wave behaviors of Kundu–Mukherjee–Naskar model arising in optical fiber communication systems with complex structure. Opt. Quant. Electron. 53(6), 1–11 (2021)

    Article  Google Scholar 

  • Sabatier, J., Agrawal, O.P., Tenreiro, M.J.A.: Advances in Fractional Calculus. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  • Saha Ray, S.: Fractional Calculus with Applications for Nuclear Reactor Dynamics. CRC Press (2015)

    Google Scholar 

  • Saha Ray, S.: A novel method for new solutions of time fractional dimensional nonlinear Schrödinger equation involving dual-power law nonlinearity. Int. J. Mod. Phys. B 33(24), 1950280 (2019). https://doi.org/10.1142/S0217979219502801

    Article  ADS  MATH  Google Scholar 

  • Saha Ray, S., Das, N.: New optical soliton solutions of fractional perturbed nonlinear Schrödinger equation in nanofibers. Mod. Phys. Lett. B 36(02), 2150544 (2022)

    Article  ADS  Google Scholar 

  • Scott, A.C., Chu, F.Y.F., McLaughlin, D.W.: The soliton: a new concept in applied science. Proc. IEEE 61(10), 1443–1483 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  • Tahir, M., Awan, A.U.: Optical singular and dark solitons with Biswas–Arshed model by modified simple equation method. Optik 202, 163523 (2020)

    Article  ADS  Google Scholar 

  • Tahir, M., Awan, A.U., Rehman, H.U.: Dark and singular optical solitons to the Biswas–Arshed model with Kerr and power law nonlinearity. Optik 185, 777–783 (2019)

    Article  ADS  Google Scholar 

  • Tozar, A., Tasbozan, O., Kurt, A.: Optical soliton solutions for the -dimensional resonant nonlinear Schröndinger’s equation arising in optical fibers. Opt. Quant. Electron. 53(6), 1–8 (2021)

    Article  Google Scholar 

  • Ur, R.H., Asjad, I.M., Bibi, M., Riaz, M., Akgül, A.: New soliton solutions of the 2D-chiral nonlinear Schrodinger equation using two integration schemes. Math. Methods Appl. Sci. 44(7), 5663–5682 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Yuan, Y.Q., Tian, B., Liu, L., Wu, X.Y., Sun, Y.: Solitons for the (2+1)-dimensional Konopelchenko–Dubrovsky equations. J. Math. Anal. Appl. 460(1), 476–486 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Zayed, E.M., Shohib, R.M., El-Horbaty, M.M., Biswas, A., Yıldırım, Y., Khan, S., Mallawi, F., Belic, M.R.: Optical dromions in cascaded systems with a couple of integration norms. Results Phys. 15, 102781 (2019). https://doi.org/10.1016/j.rinp.2019.102781

    Article  Google Scholar 

  • Zayed, E.M., Alngar, M.E., Biswas, A., Asma, M., Ekici, M., Alzahrani, A.K., Belic, M.R.: Optical solitons and conservation laws with generalized Kudryashov’s law of refractive index. Chaos Solitons Fractals 139, 110284 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, Q., Zhu, Q., Yu, H., Liu, Y., Wei, C., Yao, P., Bhrawy, A.H., Biswas, A.: Bright, dark and singular optical solitons in a cascaded system. Laser Phys. 25(2), 025402 (2014)

    Article  ADS  Google Scholar 

  • Zulfiqar, A., Ahmad, J.: Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method. Results Phys. 19, 103476 (2020)

    Article  Google Scholar 

  • Zulfiqar, A., Ahmad, J., Ul-Hassan, Q.M.: Analysis of some new wave solutions of fractional order generalized Pochhammer-chree equation using exp-function method. Opt. Quant. Electron. 54(11), 1–21 (2022)

    Article  Google Scholar 

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  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    Nilkanta Das & S. Saha Ray

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  1. Nilkanta Das
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  2. S. Saha Ray
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Das, N., Saha Ray, S. Dispersive optical soliton solutions of the \((2 + 1)\)-dimensional cascaded system governing by coupled nonlinear Schrödinger equation with Kerr law nonlinearity in plasma. Opt Quant Electron 55, 328 (2023). https://doi.org/10.1007/s11082-022-04285-4

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