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Painlevé Analysis and Kink-Type Solitary Waves of the Geophysical KdV Equation Involving a Source

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Abstract

The geophysical KdV equation is used to explore the propagation of oceanic waves. In the present paper, the geophysical KdV equation involving a source (The source is a polynomial of degree \(n\) in the unknown function) is formally introduced. Through the Painlevé analysis, it is shown that the geophysical KdV equation with the source is not integrable. Under some necessary conditions for integrability, several kink-type solitary waves to the special cases of the governing model when \(n = 2\) and \(n = 4\) are derived using the classical Kudryashov method.

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References

  1. Wazwaz, A.M.: New sets of solitary wave solutions to the KdV, mKdV, and the generalized KdV equations. Commun. Nonlinear Sci. Numer. Simulat. 13, 331–339 (2008)

    Article  MathSciNet  Google Scholar 

  2. Yan, Z.: New binary travelling-wave periodic solutions for the modified KdV equation. Phys. Lett. A 372, 969–977 (2008)

    Article  MathSciNet  Google Scholar 

  3. Wazwaz, A.M.: Kinks and solitons solutions for the generalized KdV equation with two power nonlinearities. Appl. Math. Comput. 183, 1181–1189 (2006)

    MathSciNet  Google Scholar 

  4. Geyer, A., Quirchmayr, R.: Shallow water equations for equatorial tsunami waves. Philos. Trans. R. Soc. A 376, 20170100 (2017)

    Article  MathSciNet  Google Scholar 

  5. Karunakar, P., Chakraverty, S.: Effect of Coriolis constant on Geophysical Korteweg-de Vries equation. J. Ocean Eng. Sci. 4, 113–121 (2019)

    Article  Google Scholar 

  6. Ak, T., Saha, A., Dhawan, S., Kara, A.H.: Investigation of Coriolis effect on oceanic flows and its bifurcation via geophysical Korteweg-de Vries equation. Numer. Methods Partial Differ. Equ. 36, 1234–1253 (2020)

    Article  MathSciNet  Google Scholar 

  7. Rizvi, S.T.R., Seadawy, A.R., Ashraf, F., Younis, M., Iqbal, H., Baleanu, D.: Lump and Interaction solutions of a geophysical Korteweg-de Vries equation. Results Phys. 19, 103661 (2020)

    Article  Google Scholar 

  8. Rizvi, S.T.R., Seadawy, A.R., Younis, M., Ali, I., Althobaiti, S., Mahmoud, S.F.: Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation. Results Phys. 23, 103999 (2021)

    Article  Google Scholar 

  9. Alharbi, A.R., Almatrafi, M.B.: Exact solitary wave and numerical solutions for geophysical KdV equation. J. King Saud Univ. Sci. 34, 102087 (2022)

    Article  Google Scholar 

  10. Hosseini, K., Akbulut, A., Baleanu, D., Salahshour, S., Mirzazadeh, M., Akinyemi, L.: The geophysical KdV equation: its solitons, complexiton, and conservation laws. Int. J. Geomath. 13, 12 (2022)

    Article  MathSciNet  Google Scholar 

  11. Kudryashov, N.A.: Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source. Appl. Math. Lett. 41, 41–45 (2015)

    Article  MathSciNet  Google Scholar 

  12. Weiss, J., Tabor, M., Carnevale, G.: The Painleve property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)

    Article  MathSciNet  Google Scholar 

  13. Ablowitz, M.J., Clarkson, P.A.: Soliton, Nonlinear Evolution Equations and Inverse Scattering, p. 516. Cambridge University Press (1991)

    Book  Google Scholar 

  14. Kudryashov, N.A.: Painlevé analysis and exact solutions of the fourth-order equation for description of nonlinear waves. Commun. Nonlinear Sci. Numer. Simulat. 28, 1–9 (2015)

    Article  Google Scholar 

  15. Kudryashov, N.A., Ivanova, Y.S.: Painleve analysis and exact solutions for the modified Korteweg-de Vries equation with polynomial source. Appl. Math. Comput. 273, 377–382 (2016)

    MathSciNet  Google Scholar 

  16. Wazwaz, A.M.: Painlevé integrability and lump solutions for two extended (3+1)- and (2+1)-dimensional Kadomtsev–Petviashvili equations. Nonlinear Dyn. (2022). https://doi.org/10.1007/s11071-022-08074-2

    Article  Google Scholar 

  17. Wazwaz, A.M.: Painlevé analysis for Boiti–Leon–Manna–Pempinelli equation of higher dimensions with time-dependent coefficients: Multiple soliton solutions. Phys. Lett. A 384, 126310 (2020)

    Article  MathSciNet  Google Scholar 

  18. Xu, G.Q.: Painlevé analysis, lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Appl. Math. Lett. 97, 81–87 (2019)

    Article  MathSciNet  Google Scholar 

  19. Nonlaopon, K., Mann, N., Kumar, S., Rezaei, S., Abdou, M.A.: A variety of closed-form solutions, Painlevé analysis, and solitary wave profiles for modified KdV–Zakharov–Kuznetsov equation in (3+1)-dimensions. Results Phys. 36, 105394 (2022)

    Article  Google Scholar 

  20. Singh, S., Saha Ray, S.: Painlevé analysis, auto-Bäcklund transformation and new exact solutions of (2+1) and (3+1)-dimensional extended Sakovich equation with time dependent variable coefficients in ocean physics. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.01.008

    Article  Google Scholar 

  21. Fan, L., Bao, T.: Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations. Nonlinear Dyn. 109, 3091–3100 (2022)

    Article  Google Scholar 

  22. Wazwaz, A.M., Xu, G.Q.: An extended modified KdV equation and its Painlevé integrability. Nonlinear Dyn. 86, 1455–1460 (2016)

    Article  Google Scholar 

  23. Kumar, S.: Painlevé analysis and invariant solutions of Vakhnenko–Parkes (VP) equation with power law nonlinearity. Nonlinear Dyn. 85, 1275–1279 (2016)

    Article  Google Scholar 

  24. Kaur, L., Wazwaz, A.M.: Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation. Nonlinear Dyn. 94, 2469–2477 (2018)

    Article  Google Scholar 

  25. Sivatharani, B., Subramanian, K., Sekar, A., Shanmugha Sundaram, S.: Painlevé integrability and multi-wave pattern for (2+1)-dimensional long wave-short wave resonance interaction system. Nonlinear Dyn. 109, 1935–1946 (2022)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Kudryashov, N.A.: Method for finding highly dispersive optical solitons of nonlinear differential equation. Optik 206, 163550 (2020)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  29. Kudryashov, N.A.: Highly dispersive optical solitons of the generalized nonlinear eighth-order Schrödinger equation. Optik 206, 164335 (2020)

    Article  Google Scholar 

  30. Hosseini, K., Matinfar, M., Mirzazadeh, M.: Soliton solutions of high-order nonlinear Schrödinger equations with different laws of nonlinearities. Regul. Chaotic Dyn. 26, 105–112 (2021)

    Article  MathSciNet  Google Scholar 

  31. Hosseini, K., Mirzazadeh, M., Baleanu, D., Raza, N., Park, C., Ahmadian, A., Salahshour, S.: The generalized complex Ginzburg–Landau model and its dark and bright soliton solutions. Eur. Phys. J. Plus 136, 709 (2021)

    Article  Google Scholar 

  32. Hosseini, K., Salahshour, S., Sadri, K., Mirzazadeh, M., Park, C., Ahmadian, A.: The (2+1)-dimensional hyperbolic nonlinear Schrödinger equation and its optical solitons. AIMS Math. 6, 9568–9581 (2021)

    Article  MathSciNet  Google Scholar 

  33. Hosseini, K., Mirzazadeh, M., Salahshour, S., Baleanu, D., Zafar, A.: Specific wave structures of a fifth-order nonlinear water wave equation. J. Ocean Eng. Sci. 7, 462–466 (2022)

    Article  Google Scholar 

  34. Hosseini, K., Hincal, E., Mirzazadeh, M., Salahshour, S., Obi, O.A., Rabiei, F.: A nonlinear Schrödinger equation including the parabolic law and its dark solitons. Optik 273, 170363 (2023)

    Article  Google Scholar 

  35. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solit. Fractals 30, 700–708 (2006)

    Article  MathSciNet  Google Scholar 

  36. Ali, A.T., Hassan, E.R.: General expa function method for nonlinear evolution equations. Appl. Math. Comput. 217, 451–459 (2010)

    MathSciNet  Google Scholar 

  37. Hosseini, K., Mirzazadeh, M., Vahidi, J., Asghari, R.: Optical wave structures to the Fokas–Lenells equation. Optik 207, 164450 (2020)

    Article  Google Scholar 

  38. Hosseini, K., Kaur, L., Mirzazadeh, M., Baskonus, H.M.: 1-soliton solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain model with the beta time derivative. Opt. Quantum Electron. 53, 125 (2021)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia, 99138, Turkey

    K. Hosseini, E. Hincal & B. Kaymakamzade

  2. Department of Mathematics, Near East University TRNC, Mersin 10, Nicosia, 99138, Turkey

    K. Hosseini, E. Hincal & B. Kaymakamzade

  3. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    K. Hosseini & D. Baleanu

  4. Faculty of Art and Science, University of Kyrenia, Kyrenia, TRNC, Mersin 10, Turkey

    K. Hosseini, E. Hincal & B. Kaymakamzade

  5. Institute of Space sciences-Subsidiary of INFLPR, Magurele-Bucharest, Romania

    D. Baleanu

  6. Department of Mathematics, Florida A&M University, Tallahassee, FL, 32307, USA

    S. Manukure

  7. Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey

    S. Salahshour

  8. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    S. Salahshour

  9. Faculty of Science and Letters, Piri Reis University, Tuzla, Istanbul, Turkey

    S. Salahshour

Authors
  1. K. Hosseini
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  2. D. Baleanu
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  3. E. Hincal
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  4. S. Manukure
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  5. S. Salahshour
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  6. B. Kaymakamzade
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Contributions

KH: Conceptualization, Software, Writing−original draft; DB: Conceptualization, Formal analysis, Writing—original draft; EH: Software, Writing—original draft; SM: Software, Writing—review & editing SS: Investigation, Writing—review & editing BK: Investigation, Writing—review & editing

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Correspondence to K. Hosseini.

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Hosseini, K., Baleanu, D., Hincal, E. et al. Painlevé Analysis and Kink-Type Solitary Waves of the Geophysical KdV Equation Involving a Source. Int. J. Appl. Comput. Math 10, 74 (2024). https://doi.org/10.1007/s40819-024-01706-8

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  • DOI: https://doi.org/10.1007/s40819-024-01706-8

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