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Soliton Solutions of Deformed Nonlinear Schrödinger Equations Using Ansatz Method

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Abstract

In this paper, the deformation of nonlinear Schrödinger (NLS) type equations, the so-called Camassa–Holm NLS (CH-NLS) equation and Camassa–Holm derivative NLS (CH-DNLS) equation are investigated to obtain the solitary waves solutions. These deformed equations are recently constructed using the Lagrangian deformation and loop algebra splittings. The solitary wave ansatz method is used to obtain the exact soliton solutions of these equations. The behaviours of solitons solutions are presented by 3D and 2D graphs.

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References

  1. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  Google Scholar 

  2. Lenells, J.: Conservation laws of the Camassa–Holm equation. J. Phys. A 38, 869 (2005)

    Article  MathSciNet  Google Scholar 

  3. Constantin, A., Gerdjikov, V., Ivanov, R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)

    Article  MathSciNet  Google Scholar 

  4. Liu, Z.: Tifei Qian Peakons of the Camassa–Holm equation. Appl. Math. Modell. 26, 473–480 (2002)

    Article  Google Scholar 

  5. Arnaudon, A.: On a deformation of the nonlinear Schrödinger equation. J. Phys. A Math. Theor. 49, 125202 (2016)

    Article  Google Scholar 

  6. Dong, P.L., Wu, Z.W., He, J.S.: Weakly integrable Camassa–Holm-type equations. Rom. J. Phys. 62, 109 (2017)

    Google Scholar 

  7. Mylonas, I.K., Ward, C.B., Kevrekidis, P.G., Rothos, V.M., Frantzeskakis, D.J.: Asymptotic Expansions and Solitons of the Camassa–Holm Nonlinear Schrodinger Equation. Phys. Lett. A. 381, 3965 (2017)

    Article  MathSciNet  Google Scholar 

  8. Guo, L.J., Ward, C.B., Mylonas, I.K., Kevrekidis, P.G.: Solitary waves of the Camassa–Holm derivative nonlinear Schrodinger equation. Roman. Rep. Phys. 72, 107 (2020)

    Google Scholar 

  9. Mathanaranjan, T.: Solitary wave solutions of the Camassa–Holm-Nonlinear Schrdinger equation. Results Phys. 19, 103549 (2020)

    Article  Google Scholar 

  10. Wazwaz, A.M.: Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE. method, Comput. Math. Appl. 50, 1685–1696 (2005)

  11. Wazwaz, A.M.: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Modelling 40, 499–508 (2004)

    Article  MathSciNet  Google Scholar 

  12. Wazwaz, A.M.: The tanh method for traveling wave solutions of nonlinear equations. Appl. Math. Comput. 154, 714–723 (2004)

    MathSciNet  MATH  Google Scholar 

  13. EL-Wakil, S.. A.., Abdou, M..A..: New exact traveling wave solutions using modified extended tanh-function method. Chaos Solitons Fractals 31, 840–852 (2007)

  14. Dai, C.Q., Zhang, J.F.: Jacobian elliptic function method for nonlinear differential difference equations. Chaos Solutions Fractals 27, 1042–1049 (2006)

    Article  MathSciNet  Google Scholar 

  15. Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305, 383–392 (2002)

    Article  MathSciNet  Google Scholar 

  16. Abdou, M.A.: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos Solitons Fractals 31, 95–104 (2007)

    Article  MathSciNet  Google Scholar 

  17. Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350, 103–109 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Aminikhad, H., Moosaei, H., Hajipour, M.: Exact solutions for nonlinear partial differential equations via Exp-function method. Numer. Methods Partial Differ. Equa. 26, 1427–1433 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Wang, M.L., Zhang, J.L., Li, X.Z.: The \((G^{\prime }/G )\)- expansion method and traveling wave solutions of nonlinear evolutions equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)

    Article  MathSciNet  Google Scholar 

  21. Zahran, E.H.M., Khater, Mostafa M. A.: Exact solution to some nonlinear evolution equations by The \((G^{\prime }/G )\)- expansion method. JÖkull journal 64, 5 (2014)

  22. Zeng, X., Yong, X.: A new mapping method and its applications to nonlinear partial differential equations. Phys. Lett. A 372, 6602–6607 (2008)

    Article  MathSciNet  Google Scholar 

  23. Zayed, E.M.E., Al-Nowehy, A.-G.: Solitons and other exact solutions for a class of nonlinear Schrödinger-type equations, Optik - Int. J. Light Electron Opt. 130, 1295–1311 (2017)

    Article  Google Scholar 

  24. Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Zayed, E.M.E., Hoda Ibrahim, S.A.: Exact solutions of nonlinear evolution equation in mathematical physics using the modified simple equation method. Chin. Phys. Lett. 29, 060201–4 (2012)

  26. Biswas, A.: 1-Soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A 372, 4601–4602 (2008)

    Article  MathSciNet  Google Scholar 

  27. Sassaman, R., Biswas, A.: Topological and non-topological solitons of the Klein-Gordon equations in 1+ 2 dimensions. Nonlinear Dyn. 61(1–2), 23–28 (2010)

    Article  MathSciNet  Google Scholar 

  28. Mirzazadeh, M.: Soliton solutions of Davey–Stewartson equation by trial equation method and ansatz approach. Nonlinear Dyn. 82(4), 1775–1780 (2015)

    Article  MathSciNet  Google Scholar 

  29. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston (1994)

    Book  Google Scholar 

  30. Mathanaranjan, T., Himalini, K.: Analytical solutions of the time-fractional non-linear Schrodinger equation with zero and non zero trapping potential through the Sumudu Decomposition method. J Sci Univ Kelaniya 12, 21–33 (2019)

    Article  Google Scholar 

  31. Mathanaranjan, T., Vijayakumar, D.: Laplace Decomposition Method for Time-Fractional Fornberg–Whitham Type Equations. J. Appl. Math. Phys. 9, 260–271 (2021)

    Article  Google Scholar 

  32. Yildirim, Y., Yasar, E., Adem, A.R.: A multiple exp-function method for the three model equations of shallow water waves. Nonlinear Dyn 89, 2291–2297 (2017)

    Article  MathSciNet  Google Scholar 

  33. Adem, A.R.: The generalized (1+1)-dimensional and (2+1)-dimensional Ito equations: Multiple exp-function algorithm and multiple wave solutions. Comput. Math. Appl. 71(6), 1248–1258 (2016)

    Article  MathSciNet  Google Scholar 

  34. Adem, A.R.: A (2 + 1)-dimensional Korteweg-de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws. Int. J. Mod. Phys. B 30(28), 1640001 (2016)

    Article  MathSciNet  Google Scholar 

  35. Chatibi, Y., Kinani, E. H., El., Ouhadan, A.: Lie symmetry analysis of conformable differential equations. AIMS Math. 4(4), 1133–1144 (2019)

  36. Chatibi, Y., El Kinani, E. H., Ouhadan, A.: On the discrete symmetry analysis of some classical and fractional differential equations. Math. Methods Appl. Sci., 1–11 (2019)

  37. Chatibi, Y., Kinani, E. H. El., Ouhadan, A.: Lie symmetry analysis and conservation laws for the time fractional Black-Scholes equation. Int. J. Geometr. Methods Mod. Phys. 17(01), 2050010 (2020)

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  1. Department of Mathematics and Statistics, University of Jaffna, Jaffna, Sri Lanka

    Thilagarajah Mathanaranjan

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Mathanaranjan, T. Soliton Solutions of Deformed Nonlinear Schrödinger Equations Using Ansatz Method. Int. J. Appl. Comput. Math 7, 159 (2021). https://doi.org/10.1007/s40819-021-01099-y

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