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Realization of optical solitons from nonlinear Schrödinger equation using modified Sardar sub-equation technique

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Abstract

This study presents a novel modification of the Sardar sub-equation method for solving the nonlinear SchrÖdinger equation (NLSE) with second order spatiotemporal dispersion and group velocity dispersion, which is used to describe and model the propagation of optical solitons in nonlinear media. The modification is based on introducing a new function that is used to approximate the solution of the equation. By applying this modified method, we are able to obtain exact analytical solutions for the NLSE with several classes of optical soliton solutions. The method is tested on a variety of nonlinear optical systems and is shown to be highly effective in producing accurate solutions. The results of this study demonstrate the potential of this novel approach for solving the NLSE in the context of optical solitons. These soliton solutions are of great importance in the field of science, physics, mathematics, and engineering.

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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. Scripta 95(6), 065212 (2020)

    Google Scholar 

  • Agrawal, G. P.: Nonlinear fiber optics. In Nonlinear Science at the Dawn of the 21st Century. Springer, Berlin, Heidelberg (2000)

  • Akinyemi, L.: Two improved techniques for the perturbed nonlinear Biswas-Milovic equation and its optical solitons. Optik 243, 167477 (2021)

    ADS  Google Scholar 

  • Akinyemi, L., Akpan, U., Veeresha, P., Rezazadeh, H., Inc, M.: Computational techniques to study the dynamics of generalized unstable nonlinear SchrÖdinger equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.02.011

    Article  Google Scholar 

  • Bar-Sinai, Y., et al.: Learning data-driven discretizations for partial differential equations. Proc. Nat. Acad. Sci. 116(31), 15344–15349 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  • Baskonus, H.M., Gao, W., Rezazadeh, H., Mirhosseini-Alizamini, S.M., Baili, J., Ahmad, H., Gia, T.N.: New classifications of nonlinear SchrÖdinger model with group velocity dispersion via new extended method. Results Phys. 31, 104910 (2021)

    Google Scholar 

  • Biswas, A., Fessak, M., Johnson, S., Beatrice, S., Milovic, D., Jovanoski, Z., Majid, F.: Optical soliton perturbation in non-Kerr law media. Traveling wave solution. Opt. Laser Technol. 44(1), 263–268 (2012)

    ADS  Google Scholar 

  • Chen, Y.X., Xiao, X.: Vector bright-dark one-soliton and two-soliton of the coupled NLS model with the partially nonlocal nonlinearity in BEC. Optik 257, 168708 (2022)

    ADS  Google Scholar 

  • Christian, J.M., McDonald, G.S., Hodgkinson, T.F., Chamorro-Posada, P.: Wave envelopes with second-order spatiotemporal dispersion I Bright Kerr solitons and cnoidal waves. Phys. Rev. A 86, 023838 (2012)

    ADS  Google Scholar 

  • Dai, C.Q., Wang, Y.Y.: Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals. Nonlinear Dyn. 102(3), 1733–1741 (2020)

    Google Scholar 

  • Ege, S.M., Misirli, E.: The modified Kudryashov method for solving some fractional-order nonlinear equations. Adv. Differ Equ. 2014(1), 1–13 (2014)

    MathSciNet  MATH  Google Scholar 

  • Fang, J.J., Mou, D.S., Zhang, H.C., Wang, Y.Y.: Discrete fractional soliton dynamics of the fractional Ablowitz-Ladik model. Optik 228, 166186 (2021)

    ADS  Google Scholar 

  • Fang, Y., Wu, G.Z., Wen, X.K., Wang, Y.Y., Dai, C.Q.: Predicting certain vector optical solitons via the conservation-law deep-learning method. Opt. Laser Technol. 155, 108428 (2022)

    Google Scholar 

  • Ghanbari, B., Gomez-Aguilar, J.F.: New exact optical soliton solutions for nonlinear SchrÖdinger equation with second-order spatio-temporal dispersion involving M-derivative. Mod. Phys. Lett. B 33(20), 1950235 (2019)

    ADS  Google Scholar 

  • Ghayad, M.S., Badra, N.M., Ahmed, H.M., Rabie, W.B.: Derivation of optical solitons and other solutions for nonlinear SchrÖdinger equation using modified extended direct algebraic method. Alex. Eng. J. 64, 801–811 (2023)

    Google Scholar 

  • Helal, M.A.: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos, Solitons & Fractals 13(9), 1917–1929 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ibrahim, S.: Solitary wave solutions for the (2 + 1) CBS equation. Adv. Differ. Equ. Control Process. 29, 117–126 (2022). https://doi.org/10.17654/0974324322036

    Article  Google Scholar 

  • Ibrahim, S.: Optical soliton solutions for the nonlinear third-order partial differential equation. Adv. Differ. Equ. Control Process. 29, 127–138 (2022). https://doi.org/10.17654/0974324322037

    Article  Google Scholar 

  • Ibrahim, S., Sulaiman, T.A., Yusuf, A., et al.: Families of optical soliton solutions for the nonlinear Hirota-Schrodinger equation. Opt. Quant Electron. 54, 722 (2022). https://doi.org/10.1007/s11082-022-04149-x

    Article  Google Scholar 

  • Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: New solitary wave solutions and conservation laws to the Kudryashov-Sinelshchikov equation. Optik 142, 665–673 (2017)

    ADS  Google Scholar 

  • Inc, M., Isa Aliyu, A., Yusuf, A., Baleanu, D.: Optical solitons, explicit solutions and modulation instability analysis with second-order spatio-temporal dispersion. Eur. Phys. J. Plus 132, 1–9 (2017)

    Google Scholar 

  • Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons to the resonance nonlinear SchrÖdinger equation by Sine-Gordon equation method. Superlattices Microstruct. 113, 541–549 (2018)

    ADS  Google Scholar 

  • Jawad, A.J.A.M., Petković, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217(2), 869–877 (2010)

    MathSciNet  MATH  Google Scholar 

  • Jiang, Y., et al.: Computational investigation on a nonlinear dispersion model with the weak non-local nonlinearity in quantum mechanics. Results Phys. 38, 105583 (2022)

    Google Scholar 

  • Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals 24(5), 1217–1231 (2005)

    ADS  MathSciNet  MATH  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)

    ADS  MathSciNet  MATH  Google Scholar 

  • Kumar, D., Seadawy, A.R., Chowdhury, R.: On new complex soliton structures of the nonlinear partial differential equation describing the pulse narrowing nonlinear transmission lines. Opt. Quant. Electron. 50, 1–14 (2018)

    Google Scholar 

  • Logan, J.D.: An introduction to nonlinear partial differential equations. John Wiley & Sons, UK (2008)

    MATH  Google Scholar 

  • Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82(6), 065003 (2010)

    ADS  MATH  Google Scholar 

  • Mirzazadeh, M., Eslami, M., Biswas, A.: 1-Soliton solution of KdV6 equation. Nonlinear Dyn. 80(1), 387–396 (2015)

    MathSciNet  MATH  Google Scholar 

  • Nestor, S., Houwe, A., Rezazadeh, H., Betchewe, G., Bekir, A., Doka, S.Y.: Chirped W-shape bright, dark and other solitons solutions of a conformable fractional nonlinear SchrÖdinger’s equation in nonlinear optics. Indian J. Phys. 96(1), 243–255 (2022)

    ADS  Google Scholar 

  • Polyanin, A.D., Zaitsev, V.F.: Handbook of nonlinear partial differential equations. CRC Press, USA (2004)

    MATH  Google Scholar 

  • Rashidi, M.M., Hamed, S.: Analytical solution of three-dimensional Navier-Stokes equations for the flow near an infinite rotating disk. Commun. Nonlinear Sci. Numer. Simul. 14(7), 2999–3006 (2009)

    ADS  MATH  Google Scholar 

  • Rezazadeh, H., Inc, M., Baleanu, D.: New solitary wave solutions for variants of (3+ 1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations. Front. Phys. 8, 332 (2020)

    Google Scholar 

  • Rezazadeh, H., Adel, W., Eslami, M., Tariq, K.U., Mirhosseini-Alizamini, S.M., Bekir, A., Chu, Y.M.: On the optical solutions to nonlinear SchrÖdinger equation with second-order spatiotemporal dispersion. Open Phys. 19(1), 111–118 (2021)

    Google Scholar 

  • Sabi’u, J., Jibril, A., Gadu, A.M.: New exact solution for the (3+1) conformable space-time fractional modified Korteweg-de-Vries equations via Sine-Cosine Method. J. Taibah Univ. Sci. 13(1), 91–95 (2019)

    Google Scholar 

  • Sabi’u, J., Tala-Tebue, E., Rezazadeh, H., Arshed, S., Bekir, A.: Optical solitons for the decoupled nonlinear SchrÖdinger equation using Jacobi elliptic approach. Commun. Theor. Phys. 73(7), 075003 (2021)

    ADS  Google Scholar 

  • Seadawy, A.R., Cheemaa, N.: Propagation of nonlinear complex waves for the coupled nonlinear SchrÖdinger Equations in two core optical fibers. Phys. A 529, 121330 (2019)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Mujahid, I., Dianchen, L.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78(11), 3620–3632 (2019)

    MathSciNet  MATH  Google Scholar 

  • Sulaiman, T.A.: Three-component coupled nonlinear SchrÖdinger equation. optical soliton and modulation instability analysis. Phys. Scripta 95(6), 065201 (2020)

    ADS  Google Scholar 

  • Sulaiman, T.A., Yusuf, A., Alshomrani, A.S., Baleanu, D.: Lump collision phenomena to a nonlinear physical model in coastal engineering. Mathematics 10(15), 2805 (2022)

    Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear SchrÖdinger equation by the first integral method. J. Math. Anal. Appl. 374(2), 549–553 (2011)

    MathSciNet  MATH  Google Scholar 

  • Tao, G., Sabi’u, J., Nestor, S., El-Shiekh, R.M., Akinyemi, L., Az-Zo’bi, E., Betchewe, G.: Dynamics of a new class of solitary wave structures in telecommunications systems via a (2+ 1)-dimensional nonlinear transmission line. Mod. Phys. Lett. B 36(19), 2150596 (2022)

    ADS  MathSciNet  Google Scholar 

  • Vitanov, N.K., Dimitrova, Z.I.: Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Commun. Nonlinear Sci. Numer. Simul. 15(10), 2836–2845 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  • Wen, X., Feng, R., Lin, J., Liu, W., Chen, F., Yang, Q.: Distorted light bullet in a tapered graded-index wave guide with PT symmetric potentials. Optik 248, 168092 (2021)

    ADS  Google Scholar 

  • Wen, X.K., Wu, G.Z., Liu, W., Dai, C.Q.: Dynamics of diverse data-driven solitons for the three-component coupled nonlinear SchrÖdinger model by the MPS-PINN method. Nonlinear Dyn. (2022). https://doi.org/10.1007/s11071-022-07583-4

    Article  Google Scholar 

  • Whitham, G.B.: Linear and nonlinear waves. John Wiley & Sons, UK (2011)

    MATH  Google Scholar 

  • Yusuf, A., Sulaiman, T.A.: Dynamics of Lump-periodic, breather and two-wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice. Commun. Nonlinear Sci. Numer. Simul. 99, 105846 (2021)

    MathSciNet  MATH  Google Scholar 

  • Zhang, H.: A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations. Chaos, Solitons & Fractals 39(3), 1020–1026 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  • Zhou, Q.: Analytical study of solitons in magneto-electro-elastic circular rod. Nonlinear Dyn. 83(3), 1403–1408 (2016)

    MathSciNet  MATH  Google Scholar 

  • Zimmerman, W.B.J.: Process modelling and simulation with finite element methods. World Scientific, Singapore (2004)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics of Education, Tishk International University-Erbil, Erbil, Kurdistan Region, Iraq

    Salisu Ibrahim & Younis A. Sabawi

  2. Department of Computer Engineering, Tishk International University-Erbil, Erbil, Kurdistan Region, Iraq

    Abubakar M. Ashir

  3. Department of Mathematics, Faculty of Science and Health, Koya University, Koya, Kurdistan Region, KOY45, Iraq

    Younis A. Sabawi

  4. Department of Mathematics, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  5. Institute of Space Sciences, Magurele, Bucharest, Romania

    Dumitru Baleanu

  6. Lebanese American University, 11022801, Beirut, Lebanon

    Dumitru Baleanu

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  1. Salisu Ibrahim
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  2. Abubakar M. Ashir
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  3. Younis A. Sabawi
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  4. Dumitru Baleanu
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SI, AMA, YS, and DB who were involved in conceptualization, formal analysis, and writing of the original draft, as well as review and editing.

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Correspondence to Salisu Ibrahim.

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Ibrahim, S., Ashir, A.M., Sabawi, Y.A. et al. Realization of optical solitons from nonlinear Schrödinger equation using modified Sardar sub-equation technique. Opt Quant Electron 55, 617 (2023). https://doi.org/10.1007/s11082-023-04776-y

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