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Probing wave dynamics in the modified fractional nonlinear Schrödinger equation: implications for ocean engineering

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Abstract

The nonlinear Schrödinger equation is used to model various phenomena, such as solitons self-focusing effects and rogue waves. In the ocean engineering, the modified nonlinear Schrödinger equation investigates the behavior of water waves, considering the complex interaction of dispersion nonlinearity, and dissipation effects. By introducing fractional derivatives to the model, the M-fractional conformable modified nonlinear Schrödinger equation allows for the investigation of fractional order effects, which can study more accurately the behavior of wave propagation in real-world ocean engineering. The novelty of our research lies in the application of of the M-fractional conformable derivative on the governed equation which represents an advancement in the existing work, which have used nonlinear Schrödinger equations without fractional derivatives. Two powerful techniques: the Jacobi elliptic function method and unified solver method are applied to attain solutions to the M-fractional modified nonlinear Schrödinger equation. The several results, including dark, bright, singular, periodic, and dark-bright soliton solutions are obtained which provide valuable insights into the complex behavior of water waves in ocean engineering. Additionally, 3D and contour graphs have been provided to visually illustrate the impact of the fractional order. We also illustrate these solutions at different values of the fractional order which explain how variations in this parameter affect wave propagation. These findings will contribute to the advancement of ocean engineering techniques, enhancing our ability to design and implement effective solutions for coastal protection, offshore structures, and marine renewable energy systems.

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References

  • Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  Google Scholar 

  • Ahmad, J., Rani, S., Turki, N.B., Shah, N.A.: Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method. Res. Phys. 52, 106761 (2023)

    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. Res. Phys. 51, 106719 (2023)

    Google Scholar 

  • Ahmad, J., Akram, S., Noor, K., Nadeem, M., Bucur, A., Alsayaad, Y.: Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber. Sci. Rep. 13(1), 10877 (2023)

    Article  CAS  PubMed  PubMed Central  ADS  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. Electronics 55(8), 685 (2023)

    Article  Google Scholar 

  • Akram, S., Ahmad, J., Turki, N.B., Shah, N.A.: On the exploration of soliton solutions of the nonlinear Manakov system and its sensitivity analysis. Res. Phys. 52, 106772 (2023)

    Google Scholar 

  • Ali, A., Ahmad, J., Javed, S.: Exact soliton solutions and stability analysis to (3+ 1)-dimensional nonlinear Schrödinger model. Alexandria Eng. J. 76, 747–756 (2023)

    Article  Google Scholar 

  • Allahyani, S.A., Rehman, H.U., Awan, A.U., Tag-ElDin, E.M., Hassan, M.U.: Diverse variety of exact solutions for nonlinear Gilson–Pickering equation. Symmetry 14(10), 2151 (2022)

    Article  ADS  Google Scholar 

  • Anderson, D.R.: Taylor’s formula and integral inequalities for conformable fractional derivatives. Contribut. Math. Eng. Honor Constant. Carathéodory, pp. 25-43 (2016)

  • Baskonus, H.M., Sulaiman, T.A., Bulut, H., Aktürk, T.: Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with d-potential. Superlattices Microstruct. 115, 19–29 (2018)

    Article  CAS  ADS  Google Scholar 

  • Bekir, A., Zahran, E.H.: New visions of the soliton solutions to the modified nonlinear Schrodinger equation. Optik 232, 166539 (2021)

    Article  ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Bilal, M., Haris, H., Waheed, A., Faheem, M.: The analysis of exact solitons solutions in monomode optical fibers to the generalized nonlinear Schrödinger system by the compatible techniques. Int. J. Math. Comput. Eng. 1, 149–170 (2023)

  • Çenesiz, Y., Kurt, A.: The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Universitatis Sapientiae, Mathematica 7(2), 130–140 (2015)

    Article  MathSciNet  Google Scholar 

  • Das, N., Ray, S.S.: Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model. Optik 287, 171060 (2023)

  • De Oliveira, E.C., Tenreiro Machado, J.A.: A review of definitions for fractional derivatives and integral. Math. Prob. Eng. 2014, 1–6 (2014)

    Article  MathSciNet  Google Scholar 

  • Gasmi, B., Ciancio, A., Moussa, A., Alhakim, L., Mati, Y.: New analytical solutions and modulation instability analysis for the nonlinear (1+ 1)-dimensional Phi-four model. Int. J. Math. Comput. Eng. 1, 79–90 (2023)

  • Iyiola, O.S., Nwaeze, E.R.: Some new results on the new conformable fractional calculus with application using D’Alambert approach. Prog. Fract. Differ. Appl 2(2), 115–122 (2016)

    Article  Google Scholar 

  • Katugampola, U.N.: A new fractional derivative with classical properties (2014). arXiv preprint arXiv:1410.6535

  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Computa. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  Google Scholar 

  • Kolokoltsov, V.N.: The probabilistic point of view on the generalized fractional partial differential equations. Fraction. Calculus Appl. Anal. 22, 543–600 (2019)

    Article  MathSciNet  Google Scholar 

  • Kong, H.Y., Guo, R.: Dynamic behaviors of novel nonlinear wave solutions for the Akbota equation. Optik 282, 170863 (2023)

    Article  ADS  Google Scholar 

  • Kudryashov, N.A.: Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations. Appl. Math. Comput. 371, 124972 (2020)

    MathSciNet  Google Scholar 

  • Kumar, A., Kumar, S.: Dynamic nature of analytical soliton solutions of the (1+ 1)-dimensional Mikhailov-Novikov-Wang equation using the unified approach. Int. J. Math. Comput. Eng. 1(2), 217–228 (2023)

  • Li, B., Chen, Y.: On exact solutions of the nonlinear Schrödinger equations in optical fiber. Chaos Solitons Fractals 21(1), 241–247 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  • Li, C., Chen, A.: Numerical methods for fractional partial differential equations. Int. J. Comput. Math. 95(6–7), 1048–1099 (2018)

    Article  MathSciNet  Google Scholar 

  • Li, X.Y., Liu, X.Y.: A hybrid kernel functions collocation approach for boundary value problems with Caputo fractional derivative. Appl. Math. Lett. 142, 108636 (2023)

    Article  MathSciNet  Google Scholar 

  • Liu, F., Zhuang, P., Liu, Q.: Numerical Methods of Fractional Partial Differential Equations and Applications (2015)

  • Mahmud, A.A., Tanriverdi, T., Muhamad, K.A.: Exact traveling wave solutions for (2+ 1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods. Int. J. Math. Comput. Eng. 1(1), 1–24 (2023)

  • Moualkia, S.: Mathematical analysis of new variant Omicron model driven by Lévy noise and with variable-order fractional derivatives. Chaos Solitons Fractals 167, 113030 (2023)

    Article  MathSciNet  Google Scholar 

  • Nandi, D.C., Ullah, M.S., Ali, M.Z.: Application of the unified method to solve the ion sound and Langmuir waves model. Heliyon, 8(10) (2022)

  • Plastino, A.R., & Tsallis, C.: Nonlinear Schrödinger equation in the presence of uniform acceleration. J. Math. Phys., 54(4) (2013)

  • Potasek, M.J., Tabor, M.: Exact solutions for an extended nonlinear Schrödinger equation. Phys. Lett. A 154(9), 449–452 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  • Rehman, H. U., Inc, M., Asjad, M. I., Habib, A., Munir, Q.: New soliton solutions for the space-time fractional modified third order Korteweg-de Vries equation. J. Ocean Eng. Sci. (2022)

  • Rehman, H.U., Iqbal, I., Hashemi, M.S., Mirzazadeh, M., Eslami, M.: Analysis of cubic-quartic-nonlinear Schrödinger’s equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques. Optik 287, 171028 (2023)

  • Rehman, H.U., Ullah, N., Imran, M.A.: Optical solitons of Biswas–Arshed equation in birefringent fibers using extended direct algebraic method. Optik 226, 165378 (2021)

    Article  ADS  Google Scholar 

  • Rehman, H.U., Seadawy, A.R., Younis, M., Yasin, S., Raza, S.T., Althobaiti, S.: Monochromatic optical beam propagation of paraxial dynamical model in Kerr media. Res. Phys. 31, 105015 (2021)

    Google Scholar 

  • Rehman, H.U., Awan, A.U., Tag-ElDin, E.M., Alhazmi, S.E., Yassen, M.F., Haider, R.: Extended hyperbolic function method for the (2+ 1)-dimensional nonlinear soliton equation. Res. Phys. 40, 105802 (2022)

    Google Scholar 

  • Rehman, H.U., Awan, A.U., Habib, A., Gamaoun, F., El Din, E.M.T., Galal, A.M.: Solitary wave solutions for a strain wave equation in a microstructured solid. Res. Phys. 39, 105755 (2022)

    Google Scholar 

  • Shahzad, M.U., Rehman, H.U., Awan, A.U., Zafar, Z., Hassan, A.M., & Iqbal, I.: Analysis of the exact solutions of nonlinear coupled Drinfeld-Sokolov-Wilson equation through φ6-model expansion method. Res. Phys., 106771 (2023)

  • Shi, D., Rehman, H.U., Iqbal, I., Vivas-Cortez, M., Saleem, M.S., Zhang, X.: Analytical study of the dynamics in the double-chain model of DNA. Res. Phys., 106787 (2023)

  • Sousa, J.V.D.C., de Oliveira, E.C.: On a new operator in fractional calculus and applications. arXiv preprint arXiv:1710.03712, 220 (2018)

  • Sousa, J.V.D.C., Oliveira, E.C.D.: Mittag–Leffler functions and the truncated V-fractional derivative. Mediterranean J. Math. 14(6), 244 (2017)

    Article  MathSciNet  Google Scholar 

  • Tahir, M., Awan, A.U., Rehman, H.U.: Optical solitons to Kundu–Eckhaus equation in birefringent fibers without four-wave mixing. Optik 199, 163297 (2019)

    Article  ADS  Google Scholar 

  • Ullah, M.S., Abdeljabbar, A., Roshid, H.O., Ali, M.Z.: Application of the unified method to solve the Biswas–Arshed model. Res. Phys. 42, 105946 (2022)

    Google Scholar 

  • Wang, K.J., Wang, G.D., Shi, F.: Diverse optical solitons to the Radhakrishnan-Kundu-Lakshmanan equation for the light pulses. J. Nonlinear Opt. Phys. Mater. 32, 2350074 (2023)

  • Wang, K.J., Wang, G.D., Shi, F.: The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets. In: COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 42(6), 1576–1593 (2023)

  • Wang, K.J.: New exact solutions of the local fractional modified equal width-Burgers equation on the Cantor sets. Fractals 31(9), 2550111 (2023)

  • Wang, K.J.: On the generalized variational principle of the fractal Gardner equation. Fractals 31(9), 2350120 (2023)

    Article  ADS  Google Scholar 

  • Wang, K.J.: Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+ 1)-dimensional integrable fourth-order equation for shallow water waves. Int. J. Numer. Methods Heat Fluid Flow 33(11), 734–3747(2023)

    Article  Google Scholar 

  • Wang, K.J.: Resonant multiple wave, periodic wave and interaction solutions of the new extended (3+ 1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 111(17), 16427–16439 (2023)

    Article  Google Scholar 

  • Wang, K.J.: Diverse wave structures to the modified Benjamin–Bona–Mahony equation in the optical illusions field. Mod. Phys. Lett. B 37(11), 2350012 (2023). https://doi.org/10.1142/S0218863523500741

    Article  MathSciNet  CAS  ADS  Google Scholar 

  • Wang, K.J., Shi, F.: A new fractal model of the convective-radiative fins with temperature-dependent thermal conductivity. Thermal Sci 00, 207–207 (2022)

    Google Scholar 

  • Wang, K.J., Xu, P.: Generalized variational structure of the fractal modified KdV-Zakharov-Kuznetsov equation. Fractals 31(07), 2350084 (2023)

    Article  ADS  Google Scholar 

  • Wang, K.J., Xu, P., Shi, F.: Nonlinear dynamic behaviors of the fractional (3+ 1)-dimensional modified Zakharov–Kuznetsov equation. Fractals 31(07), 2350088 (2023)

    Article  ADS  Google Scholar 

  • Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Li, L.Q.: Modified generalized Darboux transformation and solitons for a Lakshmanan–Porsezian–Daniel equation. Chaos Solitons Fractals 162, 112399 (2022)

    Article  MathSciNet  Google Scholar 

  • Yan, Z.: Generalized method and its application in the higher-order nonlinear Schrödinger equation in nonlinear optical fibres. Chaos Solitons Fractals 16(5), 759–766 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  • Yang, D., Lou, Q., Zhang, J.: Bifurcations and exact soliton solutions for generalized Dullin–Gottwald–Holm equation with cubic power law nonlinearity. Eur. Phys. J. Plus 137(2), 1–14 (2022)

    Article  Google Scholar 

  • Younis, M., Rehman, H.U., Rizvi, S.T.R., Mahmood, S.A.: Dark and singular optical solitons perturbation with fractional temporal evolution. Superlattices Microstruct. 104, 525–531 (2017)

    Article  CAS  ADS  Google Scholar 

  • Zulfiqar, A., Ahmad, J., Rani, A., Ul Hassan, Q.M.: Wave propagations in nonlinear low-pass electrical transmission lines through optical fiber medium. Math. Prob. Eng. 2022, 1–16 (2022)

    Article  Google Scholar 

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Acknowledgements

All the authors are grateful to their respective institutes for providing continuous encouragement and a conducive research environment that greatly facilitated the completion of this work.

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Authors and Affiliations

  1. Department of Biomedical Engineering, National Cheng Kung University, Tainan, 701401, Taiwan

    Dean Chou

  2. Miin Wu School of Computing, National Cheng Kung University, Tainan, 701401, Taiwan

    Dean Chou

  3. Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan, 701401, Taiwan

    Dean Chou

  4. National Center for High-performance Computing, Hsinchu, 300092, Taiwan

    Dean Chou

  5. Department of Mathematics, College of Sciences and Arts in ArRass, Qassim University, Buraydah, 51452, Saudi Arabia

    Salah Mahmoud Boulaaras

  6. Department of Mathematics, University of Okara, Okara, Pakistan

    Hamood Ur Rehman & Ifrah Iqbal

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  1. Dean Chou
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  2. Salah Mahmoud Boulaaras
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  3. Hamood Ur Rehman
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  4. Ifrah Iqbal
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Correspondence to Hamood Ur Rehman.

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Chou, D., Boulaaras, S.M., Rehman, H.U. et al. Probing wave dynamics in the modified fractional nonlinear Schrödinger equation: implications for ocean engineering. Opt Quant Electron 56, 228 (2024). https://doi.org/10.1007/s11082-023-05954-8

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