Advertisement

Some Exact Solutions of the Kudryashov–Sinelshchikov Equation Using Point Transformations

  • A. H. Abdel KaderEmail author
  • M. S. Abdel Latif
  • H. M. Nour
Original Paper
  • 5 Downloads

Abstract

In this paper, using the traveling wave ansatz, the Kudryashov–Sinelshchikov equation is transformed into a nonlinear ordinary differential equation. This nonlinear ordinary differential equation is linearized to a linear differential equation using point transformations of the independent and dependent variables. New exact solutions for the Kudryashov–Sinelshchikov equation are obtained. These exact solutions are in the form of dark soliton, bright soliton and periodic solutions.

Keywords

Exact solutions Soliton solutions Kudryashov–Sinelshchikov equation Linearization of differential equations 

Notes

References

  1. 1.
    Kudryashov, N.A., Sinelshchikov, D.I.: Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer. Fluid Dyn. 45, 96–112 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ryabov, P.N.: Exact solutions of the Kudryashov–Sinelshchikov equation. Appl. Math. Comput. 217, 3585–3590 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Mirzazadeh, M., Eslami, M.: Exact solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation via the first integral method”. Nonlinear Anal. Model. Control 17, 481–488 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Coclite, G. M., Di Ruvo, L.: A singular limit problem for the Kudryashov–Sinelshchikov equation. arXiv: 1411.5033 [math.AP] (2014)Google Scholar
  5. 5.
    Kochanov, M.B., Kudryashov, N.A.: Quasi-exact solutions of the equation for description of nonlinear waves in a liquid with gas bubbles. Rep. Math. Phys. 74, 399–408 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    He, Y.: New Jacobi elliptic function solutions for the Kudryashov–Sinelshchikov equation using improved F-expansion method. Math. Probl. Eng. 2013, 104894 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    He, Y., Li, S., Long, Y.: Exact solutions of the Kudryashov–Sinelshchikov equation by modified exp-function method. Int. Math. Forum. 8, 895–902 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    He, B., Meng, Q., Zhang, J., Long, Y.: Periodic loop solutions and their limit forms for the Kudryashov–Sinelshchikov equation. Math. Probl. Eng 2012, 320163 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    He, Y., Li, S., Long, Y.: Exact solutions of the Kudryashov–Sinelshchikov equation using the multiple G′/G-expansion method. Math. Probl. Eng. 2013, 708049 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Güner, O., Bekir, A., Cevikel, A.C.: Dark soliton and periodic wave solutions of nonlinear evolution equations. Adv. Differ. Equ. 2013, 68 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    He, B., Meng, Q., Long, Y.: The bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation. Commun. Nonlinear Sci. Numer. Simul. 17, 4137–4148 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, J., Chen, G.: Exact traveling wave solutions and their bifurcations for the Kudryashov–Sinelshchikov equation. Int. J. Bifurcat. Chaos. 22, 1250118 (2012)CrossRefGoogle Scholar
  13. 13.
    Abdel Kader, A.H., Abdel Latif, M.S., Nour, H.M.: Some new exact solutions of the modified KdV equation using lie point symmetry method. Int. J. Appl. Comput. Math. 3(Suppl. (1)), S1163–S1171 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Abdel Kader, A.H., Abdel Latif, M.S.: New soliton solutions of the CH–DP equation using lie symmetry method. Mod. Phys. Lett. B 32(20), 1850234 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Abdel Kader, A.H., Abdel Latif, M.S., Bialy, F.E., Elsaid, A.: Symmetry analysis and some new exact solutions of some nonlinear KdV-like equations. Asian Eur. J. Math. 11(3), 1850040 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Inc, M., Aliyu, A.I., Yisuf, A., Baleanu, D.: Combined optical solitary waves and conservation laws for nonlinear Chen–Lee–Liu equation in optical fibers. Optik 158, 665–673 (2018)Google Scholar
  17. 17.
    Inc, M., Hashemi, M.S., Aliyu, A.I.: Exact solutions and conservation laws of the Bogoyavlenskii equation. Acta Phys. Pol., A 13, 1133–1137 (2018)CrossRefGoogle Scholar
  18. 18.
    Inc, M., Aliyu, A.I., Yisuf, A.: Optical solitons to the nonlinear Shrödinger’s equation with spatio-temporal dispersion using complex amplitude ansatz. J. Mod. Opt. 64, 2273–2280 (2018)CrossRefGoogle Scholar
  19. 19.
    Abdel Latif, M.S.: Bright and dark soliton solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law and non-Kerr law nonlinearity. Appl. Math. Comput. 247, 501–510 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Shamseldeen, S., Abdel Latif, M.S., Hamed, A.A., Nour, H.M.: New doubly-periodic solutions for the new integrable non local modified KdV equation. J. Ocean Eng. Sci. 2, 245–247 (2017)CrossRefGoogle Scholar
  21. 21.
    Lie, S.: Classifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y die eine Gruppevon Transformationen gestatten. Math. Ann. 32, 213–281 (1888)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nakpim, W., Meleshko, S.: Linearization of second-order ordinary differential equations by generalized Sundman transformations. SIGMA 6, 051 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ibragimov, N.H., Magri, F.: Geometric proof of Lie’s linearization theorem. Nonlinear Dyn. 36, 41–46 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhao, Y.M.: F-expansion method and its application for finding new exact solutions to the Kudryashov–Sinelshchikov equation. J. Appl. Math. 2013, Article ID 895760 (2013)Google Scholar
  25. 25.
    Inc, M., Aliyu, A.I., Yisuf, A., Baleanu, D.: New solitary wave solutions and conservation laws to the Kudryashov–Sinelshchikov equation. Optik 142, 665–673 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Mathematics and Engineering Physics Department, Faculty of EngineeringMansoura UniversityMansouraEgypt

Personalised recommendations