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New soliton solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation with the beta-derivative

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Abstract

In this paper, the modified exponential function method is applied to find the exact solutions of the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative. For this, the definition of the conformable beta derivative proposed by Atangana and the properties of this derivative are firstly given. Then, the exact solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation which can be stated with the conformable beta-derivative of Atangana are obtained by using of the presented method. For the related problem in this paper, two solution cases are obtained, in each case five different solution families. The exact solutions found as a result of the application of the method seem to be 1-soliton solutions, dark soliton solutions, periodic soliton solutions and rational function solutions. According to the obtained results, it can be said that the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative has different kinds of soliton solutions. Also, three-dimensional contour and density graphs and two-dimensional graphs drawn with different parameters are given of these new exact solutions. These graphs give detailed informations about the physical behavior of the real and imaginary parts of the exact solutions obtained.

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References

  • Akturk, T.: Modified exponential function method for nonlinear mathematical models with Atangana conformable derivative. Rev. Mex. De Fis. 67(040704), 1–18 (2021)

    Google Scholar 

  • Atangana, A.: A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women. Neural Comput. Appl. 26(8), 1895–1903 (2015)

    Article  Google Scholar 

  • Atangana, A., Baleanu, D.: Theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)

    Article  Google Scholar 

  • Atangana, A., Goufo, E.F.D.: On the mathematical analysis of ebola hemorrhagic fever: deathly infection disease in West African countries. Biomed Res. Int. 261383, 1–7 (2014)

    Google Scholar 

  • Atangana, A., Oukouomi Noutchie, S.C.: Model of breakbone fever via beta-derivatives. Biomed. Res. Int. (2014). https://doi.org/10.1155/2014/523159

    Article  MATH  Google Scholar 

  • Atangana, A., Baleanu, D., Alsaedi, A.: Analysis of time fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Phys. 14, 145–149 (2016)

    Article  Google Scholar 

  • Caputo, M., Fabrizio, M.: Damage and fatigue described by a fractional derivative model. J. Comput. Phys. 293, 400–408 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  • Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, 73–85 (2015)

    Google Scholar 

  • Chen, C., Jiang, Y.L.: Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput. Math. Appl. 75, 2978–2988 (2018)

    Article  MathSciNet  Google Scholar 

  • Darvishi, M.T., Najafi, M., Seadawy, A.R.: Dispersive bright, dark and singular optical soliton solutions in conformable fractional optical fiber Schrödinger models and its applications. Opt. Quantum Electron. 50(181), 1–16 (2018)

    Google Scholar 

  • Das, N., Singh, R., Wazwaz, A.M., Kumar, J.: An algorithm based on the variational iteration technique for the Bratu-type and the Lane-Emden problem J. . Math. Chem. 54, 527–551 (2016)

    Article  MathSciNet  Google Scholar 

  • Demiray, S.T.: New solutions of Biswas-Arshed equation with beta time derivative. Optik 222, 165405 (2020)

    Article  ADS  Google Scholar 

  • Demiray, S.T., Pandir, Y., Bulut, H.: The investigation of exact solutions of nonlinear time-fractional Klein-Gordon equation by using generalized Kudryashov method. AIP Conf. Proc. 1637, 283–289 (2014b)

    Article  ADS  Google Scholar 

  • Tuluce Demiray, S., Pandir, Y., Bulut, H. Generalized Kudryashov method for time-fractional differential equations, Abstr. Appl. Anal. 1–13 (2014a)

  • Gao, G.H., Sun, Z.Z., Zhang, Y.N.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231(7), 2865–2879 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  • Ghanbari, B., Gomez-Aguilar, J.F.: The generalized exponential rational function method for Radhakrishnan-Kundu Lakshmanan equation with β-conformable time derivative. Rev. Mex. De Fis. 65(5), 503–518 (2019)

    Article  MathSciNet  Google Scholar 

  • Guo, S., Mei, L., Li, Y., Sun, Y.: The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Phys. Lett. A 376, 407–411 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  • Gurefe, Y.: The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative. Rev. Mex. De Fis. 66(6), 771–781 (2020)

    Article  MathSciNet  Google Scholar 

  • Hosseini, K., Mirzazadeh, M., Gomez-Aguilar, J.F.: Soliton solutions of the Sasa-Satsuma equation in the monomode optical fibers including the beta-derivatives. Optik 224, 165425 (2020)

    Article  ADS  Google Scholar 

  • Hosseini, K., Mirzazadeh, M., Ilie, M., Gomez-Aguilar, J.F.: Biswas-Arshed equation with the beta time derivative: optical solitons and other solutions. Optik 217, 164801 (2020)

    Article  ADS  Google Scholar 

  • Hu, M.S., Agarwal, R.P., Yang, X.J.: Local fractional Fourier series with application to wave equation in fractal vibrating string, Abstr. Appl. Anal. 2012 (2012) 567401.

  • Jafari, H., Jassim, H.K.: Numerical solutions of telegraph and Laplace equations on cantor sets using local fractional Laplace decomposition method. Int. J. Adv. Appl. Math. Mech. 2(3), 144–151 (2015)

    MathSciNet  MATH  Google Scholar 

  • Jumarie, G.: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51(9–10), 1367–1376 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Kumar, D., Singh, J., Kumar, S.: Numerical computation of fractional Black-Scholes equation arising in financial market. J. Egypt. Math. Soc. 22, 373–378 (2014)

    Article  Google Scholar 

  • Lu, B.: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395(2), 684–693 (2012)

    Article  MathSciNet  Google Scholar 

  • Pandir, Y., Gurefe, Y., Misirli, E.: The extended trial equation method for some time-fractional differential equations. Discrete Dyn Nat Soc 491359, 13 (2013)

    MathSciNet  MATH  Google Scholar 

  • Pandir, Y., Gurefe, Y., Misirli, E.: New exact solutions of the time-fractional nonlinear dispersive KdV equation. Int. J. Model. Optim. 3, 349–352 (2013c)

    Article  Google Scholar 

  • Pandir ,Y., Gurefe Y., Misirli, E. The extended trial equation method for some time-fractional differential equations, Discrete Dyn. Nat. Soc. 2 491359 (2013b)

  • Podlubny, I.: Fractional Differential Equations, (Academic Press, 1999).

  • Rahman, M.A.: The exp (−Φ (η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys. 4, 150–155 (2014)

    Article  Google Scholar 

  • Rani, D., Mishra, V.: Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson formula. Eur. J. Pure Appl. Math. 11, 202–214 (2018)

    Article  MathSciNet  Google Scholar 

  • Tuluce Demiray, S., Bayrakci, U.: Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified exp(−Ω(η))-expansion function method. Rev. Mex. de Fis. 67(3), 393–402 (2021)

    Google Scholar 

  • Yepez-Martinez, H., Gomez-Aguilar, J.F.: Fractional sub-equation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative. Discrete Dyn. Nat. Soc. 29(4), 678–693 (2019)

    MathSciNet  Google Scholar 

  • Yepez-Martinez, H., Gomez-Aguilar, J.F.: Optical solitons solution of resonance nonlinear Schrodinger type equation with Atangana’s conformable derivative using sub-equation method. Discrete Dyn. Nat. Soc. 31(3), 573–596 (2021a)

    MathSciNet  Google Scholar 

  • Yepez-Martinez, H., Gomez-Aguilar, J.F.: Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana’s-conformable derivative using sub-equation method. Waves Random Complex Media 31(3), 573–596 (2021b)

    Article  ADS  MathSciNet  Google Scholar 

  • Yepez-Martinez, H., Gomez-Aguilar, J.F., Atangana, A.: First integral method for non-linear differential equations with conformable derivative. Math. Model. Nat. Phenom. 13(1), 1–22 (2018a)

    Article  MathSciNet  Google Scholar 

  • Yepez-Martinez, H., Gomez-Aguilar, J.F., Baleanu, D.: Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Optik 155, 357–365 (2018b)

    Article  ADS  Google Scholar 

  • Zhang, S., Zhang, H.Q.: Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 375, 1069–1073 (2011)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Yozgat Bozok University, 66100, Yozgat, Turkey

    Yusuf Pandir

  2. Department of Mathematics, Faculty of Arts and Sciences, Mersin University, 33343, Mersin, Turkey

    Yusuf Gurefe

  3. Department of Mathematics and Science Education, Faculty of Education, Ordu University, 52200, Ordu, Turkey

    Tolga Akturk

Authors
  1. Yusuf Pandir
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  2. Yusuf Gurefe
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  3. Tolga Akturk
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Correspondence to Yusuf Pandir.

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Pandir, Y., Gurefe, Y. & Akturk, T. New soliton solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation with the beta-derivative. Opt Quant Electron 54, 216 (2022). https://doi.org/10.1007/s11082-022-03585-z

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