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Discovering new abundant optical solutions for the resonant nonlinear Schrödinger equation using an analytical technique

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Abstract

In our exploration of optical physics, the intricate resonant nonlinear Schrödinger (NLS) equation featuring dual-power law nonlinearity is investigated which is an equation of paramount importance in the field of optics. This equation serves as a key to unlocking the intricacies of optical phenomena, including solitons, nonlinear effects, and wave interactions. Various optical solutions covering a broad variety of mathematical expressions, from trigonometric and hyperbolic functions to rational ones, are revealed by applying the technique of the powerful (\(\dot{G}/G\),\(1/G\))-expansion analytical approach which is the main goal of this study. The utmost precision and reliability of our findings are rigorously confirmed via the robust Mathematica software. Furthermore, the dynamic visual representations including 2D, 3D, and contour charts are presented to vividly depict various optical patterns such as single periodic, multi-periodic, singular soliton, and semi-bell-shaped phenomena. These solutions are of the utmost significance in the fields of nonlinear fiber optics and telecommunications, contributing to our comprehension of the fundamental physical concepts underlying the equation. The adaptability and application of our new and standardized technique is demonstrated by applying it to a wide range of mathematical and physical challenges.

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References

  • Abdel-Gawad, H.I., Osman, M.S.: On the variational approach for analyzing the stability of solutions of evolution equations. Kyungpook Math. J. 53(4), 661–680 (2013)

    MathSciNet  Google Scholar 

  • Abdel-Gawad, H.I., Osman, M.: On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients. J. Adv. Res. 6, 593–599 (2015)

    Google Scholar 

  • Ahmad, J., Akram, S., Rehman, S.U., Turki, N.B., Shah, N.A.: Description of soliton and lump solutions to M-truncated stochastic Biswas-Arshed model in optical communication. Result Phys. 51, 106719 (2023)

    Google Scholar 

  • Akram, S., Ahmad, J., Rehman, S.U.: Stability analysis and dynamical behavior of solitons in nonlinear optics modelled by Lakshmanan–Porsezian–Daniel equation. Opt. Quant. Electron. 55, 685 (2023)

    Google Scholar 

  • Babajanov, B., Abdikarimov, F.: The application of the functional variable method for solving the loaded non-linear evaluation equations. Front. Appl. Math Stat. 8, 1–9 (2022)

    Google Scholar 

  • Baskonus, H.M., Osman, M.S., Rehman, H., Ramzan, M., Tahir, M., Ashraf, S.: On pulse propagation of soliton wave solutions related to the perturbed Chen–Lee–Liu equation in an optical fiber. Opt. Quantum Electron. 53(10), 556 (2021)

    Google Scholar 

  • Bekir, A., San, S.: The functional variable method to some complex nonlinear evolution equations. J. Mod. Math. Front. 1(3), 5–9 (2012)

    Google Scholar 

  • Boakye, G., Hosseini, K., Hinçal, E., Sirisubtawee, S., Osman, M.S.: Some models of solitary wave propagation in optical fibers involving Kerr and parabolic laws. Opt. Quant. Electron. 56(3), 345 (2024)

    Google Scholar 

  • Chowdhury, M.A., Miah, M.M., Iqbal, M.A., Alshehri, H.M., Baleanu, D., Osman, M.S.: Advanced exact solutions to the nano-ionic currents equation through MTs and the soliton equation containing the RLC transmission line. Eur. Phys. J. plus 138, 502 (2023)

    Google Scholar 

  • Elsayed, E.M.E., Alurrfi, K.A.E.: The generalized projective Riccati equations method and its applications for solving two nonlinear PDEs describing microtubules. Int. J. Phys. Sci. 10, 391–402 (2015)

    Google Scholar 

  • El-Sherif, A.A., Shoukry, M.M.: Copper (II) complexes of imino-bis (methyl phosphonic acid) with some bio-relevant ligands. Equilibrium studies and hydrolysis of glycine methyl ester through complex formation. J. Coord. Chem. 58(16), 1401–1415 (2005)

    Google Scholar 

  • Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)

    ADS  MathSciNet  Google Scholar 

  • Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403–406 (1998)

    ADS  Google Scholar 

  • Fetoh, A., Asla, K.A., El-Sherif, A.A., El-Didamony, H., El-Reash, G.M.A.: Synthesis, structural characterization, thermogravimetric, molecular modelling and biological studies of Co (II) and Ni (II) Schiff bases complexes. J. Mol. Struct. 1178, 524–537 (2019)

    ADS  Google Scholar 

  • Fokas, A.S., Lenells, J.: The unified method: I Nonlinearizable problems on the half-line. J. Phys. A Math. Theor. 45(19), 195201 (2012)

    ADS  MathSciNet  Google Scholar 

  • Ganie, A.H., Sadek, L.H., Tharwat, M.M., Iqbal, M.A., Miah, M.M., Rasid, M.M., Elazab, N.S., Osman, M.M.: New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering. Part. Differ. Eqn. Appl. Math. 9, 100608 (2024)

    Google Scholar 

  • Habib, M.A., Ali, H.M.S., Miah, M.M., Akbar, M.A.: The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs. AIMS Math. 4, 896–909 (2019)

    MathSciNet  Google Scholar 

  • Hosseini, K., Alizadeh, F., Hinçal, E., Baleanu, D., Akgül, A., Hassan, A.M.: Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation. Result Phys. 54, 107129 (2023a)

    Google Scholar 

  • Hosseini, K., Sadri, K., Hinçal, E., Sirisubtawee, S., Mirzazadeh, M.: A generalized nonlinear Schrödinger involving the weak nonlocality: its Jacobi elliptic function solutions and modulational instability. Optik 288, 171176 (2023b)

    ADS  Google Scholar 

  • Hosseini, K., Hinçal, E., Ilie, M.: Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation. Nonlinear Dyn. 111(18), 17455–17462 (2023c)

    Google Scholar 

  • Iqbal, M.A., Baleanu, D., Miah, M.M., Ali, H.M.S., Alshehri, H.M., Osman, M.S.: New soliton solutions of the mZK equation and the Gerdjikov-Ivanov equation by employing the double (G′/G, 1/G)-expansion method. Result Phys. 47, 106391 (2023)

    Google Scholar 

  • Irshad, A., Mohyud-din, S.T., Ahmed, N., Khan, U.: A new modification in simple equation method and its applications on nonlinear equations of physical nature. Result Phys. 7, 4232–4240 (2017)

    ADS  Google Scholar 

  • Islam, S., Khan, K., Arnous, A.H.: Generalized Kudryashov method for solving some (3+ 1)-dimensional nonlinear evolution equations. New Trends Math. Sci. 57, 46–57 (2015)

    MathSciNet  Google Scholar 

  • Islam, S.M.R., Arafat, S.M.Y., Wang, H.: Abundant closed-form wave solutions to the simplified modified Camassa–Holm equation. J. Ocean Eng. Sci. 8, 238–245 (2023)

    Google Scholar 

  • Islam, M.N., Al-Amin, M., Akbar, A., Wazwaz, A.M., Osman, M.S.: Assorted optical soliton solutions of the nonlinear fractional model in optical fibers possessing beta derivative. Physica Scr. 99(1), 015227 (2024)

    ADS  Google Scholar 

  • Jafari, H., Kadkhoda, N., Baleanu, D.: Fractional Lie group method of the time-fractional Boussinesq equation. Nonlinear Dyn. 81, 1569–1574 (2015)

    MathSciNet  Google Scholar 

  • Kaur, L.: Generalized (G’/G)-expansion method for generalized fifth order KdV equation with time-dependent coefficients. Math. Sci. Lett. 3, 255–261 (2014)

    Google Scholar 

  • Kumar, A., Pankaj, R.D.: Tanh–coth scheme for traveling wave solutions for Nonlinear Wave Interaction model. J. Egypt. Math. Soc. 23, 282–285 (2015)

    MathSciNet  Google Scholar 

  • Kumar, D., Park, C., Tamanna, N., Paul, G.C., Osman, M.S.: Dynamics of two-mode Sawada-Kotera equation: mathematical and graphical analysis of its dual-wave solutions. Result Phys. 19, 103581 (2020)

    Google Scholar 

  • Kumar, D., Paul, G.C., Seadawy, A.R., Darvishi, M.T.: A variety of novel closed-form soliton solutions to the family of Boussinesq-like equations with different types. J. Ocean Eng. Sci. 7, 543–554 (2022)

    Google Scholar 

  • Ma, W.X.: AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials. Appl. Math. Lett. 145, 108775 (2023)

    MathSciNet  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  Google Scholar 

  • Mamun, A.A., Ananna, S.N., An, T., Asaduzzaman, M., Miah, M.M.: Solitary wave structures of a family of 3D fractional WBBM equation via the tanh–coth approach. Partial Differ. Equ. Appl. Math. 5, 100237 (2022)

    Google Scholar 

  • Mirzazadeh, M.: Topological and non-topological soliton solutions of Hamiltonian amplitude equation by He’s semi-inverse method and ansatz approach. J. Egypt. Math. Soc. 23, 292–296 (2015)

    MathSciNet  Google Scholar 

  • Mohanty, S.K., Kravchenko, O.V., Deka, M.K., Dev, A.N., Churikov, D.V.: The exact solutions of the 2+ 1–dimensional Kadomtsev-Petviashvili equation with variable coefficients by extended generalized G′ G-expansion method. J. King Saud Univ. - Sci. 35, 102358 (2023)

    Google Scholar 

  • Naher, H., Abdullah, F.A.: The basic (G’/G)-expansion method for the fourth order Boussinesq equation. Appl. Math. 03, 1144–1152 (2012)

    Google Scholar 

  • Parkes, E.J., Duffy, B.R.: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 98, 288–300 (1996)

    ADS  Google Scholar 

  • Rehman, S.U., Ahmad, J.: Stability analysis and novel optical pulses to Kundu–Mukherjee–Naskar model in birefringent fibers. Int. J. Mod. Phys. B (2023). https://doi.org/10.1142/S0217979224501923

    Article  Google Scholar 

  • Rehman, S.U., Bilal, M., Ahmad, J.: Highly dispersive optical and other soliton solutions to fiber Bragg gratings with the application of different mechanisms. Int. J. Mod. Phys. B 36(28), 2250193 (2022a)

    ADS  Google Scholar 

  • Rehman, S.U., Bilal, M., Ahmad, J.: The study of solitary wave solutions to the time conformable Schrödinger system by a powerful computational technique. Opt. Quant. Electron. 54, 228 (2022b)

    Google Scholar 

  • Rehman, S.U., Ahmad, J., Muhammad, T.: Dynamics of novel exact soliton solutions to Stochastic Chiral Nonlinear Schrödinger Equation. Alex. Eng. J. 79, 568–580 (2023)

    Google Scholar 

  • Roshid, H.O., Kabir, M.R., Bhowmik, R.C., Datta, B.K.: Investigation of Solitary wave solutions for Vakhnenko–Parkes equation via exp-function and Exp (− ϕ (ξ))-expansion method. Springerplus 3, 1–10 (2014)

    Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M.: The first integral method to some complex nonlinear partial differential equations. J. Comput. Appl. Math. 235, 4871–4877 (2011)

    MathSciNet  Google Scholar 

  • Tandel, P., Patel, H., Patel, T.: Tsunami wave propagation model: A fractional approach. J. Ocean Eng. Sci. 7, 509–520 (2022)

    Google Scholar 

  • Tozar, A., Tasbozan, O., Kurt, A.: Optical soliton solutions for the (1+ 1)-dimensional resonant nonlinear Schröndinger’s equation arising in optical fibers. Opt. Quantum Electron. 53(6), 316 (2021)

    Google Scholar 

  • Wang, M., Zhou, Y., Li, Z.: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A 216, 67–75 (1996)

    ADS  Google Scholar 

  • Wen, X., Lü, D.: Extended Jacobi elliptic function expansion method and its application to nonlinear evolution equation. Chaos Solitons Fractals 41, 1454–1458 (2009)

    ADS  MathSciNet  Google Scholar 

  • Yomba, E.: General projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations. Chin. J. Phys. 43, 991–1003 (2005)

    MathSciNet  Google Scholar 

  • Zayed, E.M.E., Alurrfi, K.A.E.: Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations. Appl. Math. Comput. 289, 111–131 (2016)

    MathSciNet  Google Scholar 

  • Zhang, Z.Y.: Exact traveling wave solutions of the perturbed Klein-Gordon equation with quadratic nonlinearity in (1+ 1)-dimension, Part I: Without local inductance and dissipation effect. Turkish J. Phys. 37, 259–267 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Dhaka University of Engineering and Technology, Gazipur, 1707, Bangladesh

    Md Nur Hossain

  2. Department of Civil Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka, Japan

    Md Nur Hossain

  3. Department of Mathematics, Khulna University of Engineering and Technology, Khulna, 9203, Bangladesh

    M. Mamun Miah

  4. Division of Mathematical and Physical Sciences, Kanazawa University, Kanazawa, 9201192, Japan

    M. Mamun Miah

  5. Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, 11673, Riyadh, Saudi Arabia

    Abdul Hamid Ganie

  6. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    M. S. Osman

  7. Mathematics Department, Faculty of Sciences, Umm AI-Qura University, Makkah, 21955, Saudi Arabia

    M. S. Osman

  8. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Wen-Xiu Ma

  9. Department of Mathematics, King Abdulaziz University, 21589, Jeddah, Saudi Arabia

    Wen-Xiu Ma

  10. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Wen-Xiu Ma

  11. Material Science Innovation and Modelling, North-West University, Mafikeng Campus, Potchefstroom, South Africa

    Wen-Xiu Ma

Authors
  1. Md Nur Hossain
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  2. M. Mamun Miah
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  3. Abdul Hamid Ganie
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  4. M. S. Osman
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  5. Wen-Xiu Ma
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Contributions

MNH and MMM: Conceptualization, Methodology, Software, Validation, Resources, Writing-original draft. AHG: Data curation, Writing-original draft. MSO and WXM: Supervision, Project administration, Funding acquisition, Writing-review editing, Formal analysis.

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Correspondence to M. Mamun Miah, M. S. Osman or Wen-Xiu Ma.

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Hossain, M.N., Miah, M.M., Ganie, A.H. et al. Discovering new abundant optical solutions for the resonant nonlinear Schrödinger equation using an analytical technique. Opt Quant Electron 56, 847 (2024). https://doi.org/10.1007/s11082-024-06351-5

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