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Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique

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Abstract

This study investigates new soliton-type solutions to the new generalized KdV (ngKdv) equation. For this purpose, the homogeneous balance method is used to create Auto-Bäcklund transformations of the regarded equation and with the help of the transformations, abundant exact and explicit solutions have been found. We found complexiton solutions to the dealt equation by using the extended transformed rational function technique. We have also given the 3D graphics of the obtained solutions.

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References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  • Arshed, S., Raza, N.: Optical solitons perturbation of Fokas-Lenells equation with full nonlinearity and dual dispersion. Chin. J. Phys. 63, 314–324 (2020)

    Article  MathSciNet  Google Scholar 

  • Arshed, S., Raza, N., Butt, A.R., Javid, A. , Gomez-Aguilar, J. F.: Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.11.001

  • Chen, Y., Yan, Z.: New exact solutions of (\(2+1\))dimensional Gardner equation via the new sine-Gordon equation expansion method. Chaos Solitons Fractals 26(2), 399–406 (2005)

    Article  MathSciNet  Google Scholar 

  • Ekici, M., Sonmezoglu, A., Biswas, A., Belic, M.R.: Optical solitons in (\(2+ 1\))Dimensions with KunduMukherjeeNaskar equation by extended trial function scheme. Chin. J. Phys. 57, 72–77 (2019)

    Article  Google Scholar 

  • Ganji, Z.Z., Ganji, D.D., Bararnia, H.: Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl. Math. Model. 33(4), 1836–1841 (2009)

    Article  MathSciNet  Google Scholar 

  • Hirota, R.: Exact solution of the Korteweg-de vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  ADS  Google Scholar 

  • Ismael, H.F., Murad, M. A. S., Bulut, H.: Various exact wave solutions for KdV equation with time-variable coefficients. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.11.001

  • Javeed, S., Riaz, S., Saleem Alimgeer, K., Atif, M., Hanif, A., Baleanu, D.: First integral technique for finding exact solutions of higher dimensional mathematical physics models. Symmetry 11(6), 783 (2019)

    Article  Google Scholar 

  • Kaplan, M., Akbulut, A., Bekir, A.: The auto-Bäcklund transformations for the (\(2+1\))dimensional Boiti-Leon-Manna-Pempinelli equation. In AIP Conference Proceedings 1798(1), 020071 (2017)

  • Kaplan, M., Ozer, M.N.: Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation. Opt. Quantum Electron. 50(1), 33 (2018)

    Article  Google Scholar 

  • Ma, W.X., You, Y.: Solving the KortewegdeVries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357(5), 1753–1778 (2004)

    Article  Google Scholar 

  • Ma, H., Yue, S., Deng, A.: D’Alembert wave, the Hirota conditions and soliton molecule of a new generalized KdV equation. J. Geom. Phys. 172, 104413 (2022)

    Article  MathSciNet  Google Scholar 

  • Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    Book  Google Scholar 

  • Raza, N., Rafiq, M.H., Kaplan, M., Kumar, S., Chu, Y.M.: The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations. Results Phys. 22, 103979 (2021)

    Article  Google Scholar 

  • Shang, Y.: Backlund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation. Appl. Math. Comput. 187, 1286–1297 (2007)

    MathSciNet  MATH  Google Scholar 

  • Shang, Y., Huang, Y., Yuan, W.: Backlund transformations and abundant exact explicit solutions of the SharmaTassoOlver equation. Appl. Math. Comput. 217, 7172–7183 (2011)

    MathSciNet  MATH  Google Scholar 

  • Yagmurlu, N.M., Karaagac, B., Esen, A.: A Lumped Galerkin finite element method for the generalized Hirota-Satsuma coupled KdV and coupled MKdV equations. Tbilisi Math. J. 12(3), 159–173 (2019)

    Article  MathSciNet  Google Scholar 

  • Yokus, A., Bulut, H.: Numerical simulation of KdV equation by finite difference method. Indian J. Phys. 92(12), 1571–1575 (2018)

    Article  ADS  Google Scholar 

  • Yokus, A., Baskonus, H.M., Sulaiman, T.M., Bulut, H.: Numerical simulation and solutions of the two-component second order KdV evolutionary system. Numer. Methods Partial Differ. Equ. 34(1), 211–227 (2018)

    Article  MathSciNet  Google Scholar 

  • Zhang, B., Hu, H.: Similarity reduction and exact solutions of a Boussinesq-like equation. Zeitschrift für Naturforschung A, 73(4) (2018) 357–362

  • Zhang, H., Ma, W.X.: Extended transformed rational function method and applications to complexiton solutions. Appl. Math. Comput. 230, 509–515 (2014)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan

    Nahal Jannat & Nauman Raza

  2. Department of Computer Engineering, Faculty of Engineering and Architecture, Kastamonu University, Kastamonu, Turkey

    Melike Kaplan

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  1. Nahal Jannat
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  2. Melike Kaplan
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  3. Nauman Raza
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Correspondence to Melike Kaplan.

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Jannat, N., Kaplan, M. & Raza, N. Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique. Opt Quant Electron 54, 466 (2022). https://doi.org/10.1007/s11082-022-03862-x

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