Skip to main content
SpringerLink
Log in

Lump solution and its interaction to (3+1)-D potential-YTSF equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper studies the \((3+1)\)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by implementing the Hirota bilinear method. As a consequence, the Hirota bilinear method is successfully employed and acquired a type of the lump solution and five types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions are trustworthy in the applied sciences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Roshid, H.O.: Lump solutions to a (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) like equation. Int. J. Appl. Comput. Math. 3, 1455–1461 (2017)

    Article  MathSciNet  Google Scholar 

  2. Yu, S.J., Toda, K., Sasa, N., Fukuyama, T.: N soliton solutions to the Bogoyavlenskii–Schiff equation and a quest for the soliton solution in (3+1) dimensions. J. Phys. A 31(14), 3337–3347 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yan, Z.Y.: New families of non-travelling wave solutions to a new (3+1)-dimensional potential-YTSF equation. Phys. Lett. A 318(12), 78–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, T.X., Xuan, H.N., Zhang, D., Wang, C.: Non-travelling wave solutions to a (3+1)-dimensional potential-YTSF equation and a simplified model for reacting mixtures. Chaos Solitons Frac. 34(3), 1006–1013 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zeng, X.P., Dai, Z.D., Li, D.: New periodic soliton solutions for the (3+1)-dimensional potential-YTSF equation. Chaos Solitons Frac. 42(2), 657–661 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Roshid, H.O., Akbar, M.A., Alam, M.N., Hoque, M.F., Rahman, N.: New extended (G’/G)-expansion method to solve nonlinear evolution equation: the (3+1)-dimensional potential-YTSF equation. Springer-Plus 3, 122 (2014)

    Article  Google Scholar 

  7. Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 1–20 (2015)

    Article  Google Scholar 

  8. Arnous, A.H., Seithuti, M.Z.U., Moshokoa, P., Zhou, Q., Triki, H., Mirzazadeh, M., Biswas, A.: Optical solitons in nonlinear directional couplers with trial function scheme. Nonlinear Dyn. 88, 1891–1915 (2017)

    Article  MATH  Google Scholar 

  9. Zhou, Q., Ekici, M., Sonmezoglu, A., Manafian, J., Khaleghizadeh, S., Mirzazadeh, M.: Exact solitary wave solutions to the generalized Fisher equation. Optik 127, 12085–12092 (2016)

    Article  Google Scholar 

  10. Ekici, M., Zhou, Q., Sonmezoglu, A., Manafian, J., Mirzazadeh, M.: The analytical study of solitons to the nonlinear Schrödinger equation with resonant nonlinearity. Optik 130, 378–382 (2017)

    Article  Google Scholar 

  11. Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the \(tan(\phi /2)\)-expansion method. Optik 127, 4222–4245 (2016)

    Article  Google Scholar 

  12. Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt. Quant. Elec. 48, 1–32 (2016)

    Article  Google Scholar 

  13. Sindi, C.T., Manafian, J.: Wave solutions for variants of the KdV–Burger and the K(n, n)-Burger equations by the generalized \(G^{\prime }/G\)-expansion method. Math. Meth. Appl. Sci. 87, 1–14 (2016)

    MATH  Google Scholar 

  14. Baskonus, H.M.: New complex and hyperbolic function solutions to the generalized double combined Sinh–Cosh–Gordon equation. AIP Conf. Proc. 1798, 020018 (2017)

    Article  Google Scholar 

  15. Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2+1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Random Complex Media 26, 201–208 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Zhou, Q.: Optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Waves Random Complex Media 25, 52–59 (2016)

    Article  MATH  Google Scholar 

  17. Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-Linear Mech. 31, 329–338 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ma, W.X., Lee, J.-H.: A transformed rational function method and exact solutions to the \(3+1\) dimensional Jimbo–Miwa equation. Chaos Solitons Frac. 42, 1356–1363 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010)

    Article  MATH  Google Scholar 

  20. Ma, W.X., Zhu, Z.: Solving the \((3+1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Ma, W.X.: A refined invariant subspace method and applications to evolution equations. Sci. China Math. 55, 1769–1778 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mirzazadeh, M., Eslami, M.: Exact multisoliton solutions of nonlinear Klein–Gordon equation in \(1+2\) dimensions. The Eur. Phys. J. Plus 128, 1–9 (2015)

    Google Scholar 

  23. Tchier, F., Yusuf, A., Aliyu, A.I., Inc, M.: Soliton solutions and conservation laws for lossy nonlinear transmission line equation. Superlattices Microstructures 107, 320–336 (2017)

    Article  Google Scholar 

  24. Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. 87, 1685–1691 (2017)

    Article  MathSciNet  Google Scholar 

  25. Wazwaz, A.M., El-Tantawy, S.A.: New \((3+ 1)\)-dimensional equations of Burgers type and Sharma–Tasso–Olver type: multiple-soliton solutions. Nonlinear Dyn. 87, 2457–2461 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Talati, D., Wazwaz, A.M.: Some new integrable systems of two-component fifth-order equations. Nonlinear Dyn. 87, 1111–1120 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wazwaz, A.M.: Abundant solutions of various physical features for the \((2+1)\)-dimensional modified KdV–Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. 88, 1727–1732 (2017)

    Article  MathSciNet  Google Scholar 

  28. Yu, F., Feng, L., Li, L.: Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions. Nonlinear Dyn. 88, 1257–1271 (2017)

    Article  MATH  Google Scholar 

  29. Geng, X., Lv, Y.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dyn. 69, 1621–1630 (2012)

    Article  MATH  Google Scholar 

  30. Guo, R., Tian, B., Wang, L.: Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions. Nonlinear Dyn. 88, 1257–1271 (2017)

    Article  Google Scholar 

  31. Wen, L.-L., Zhang, H.-Q.: Darboux transformation and soliton solutions of the \((2 + 1)\)-dimensional derivative nonlinear Schrödinger hierarchy. Nonlinear Dyn. 84, 863–873 (2016)

    Article  MATH  Google Scholar 

  32. Lü, X., Tian, B., Zhang, H.-Q., Xu, T., Li, H.: Generalized \((2+1)\)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms. Nonlinear Dyn. 67, 2279–2290 (2012)

    Article  MATH  Google Scholar 

  33. Na, L.: Bäcklund transformation and multi-soliton solutions for the \((3+1)\)-dimensional BKP equation with Bell polynomials and symbolic computation. Nonlinear Dyn. 82, 311–318 (2015)

    Article  MATH  Google Scholar 

  34. Lü, X., Lin, F.: Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. Numer. Simulat. 32, 241–261 (2016)

    Article  MathSciNet  Google Scholar 

  35. Lü, X., Lin, F.: Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödingerequation. Commun. Nonlinear Sci. Numer. Simulat. 31, 40–46 (2016)

    Article  Google Scholar 

  36. Kharif, C., Pelinovsky, E., Slunyaey, A.: Rogue Waves in the Ocean, Observation, Theories and Modeling. Springer, New York (2009)

    Google Scholar 

  37. Peregrine, D.H.: Water waves, nonlinear schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B Appl. Math. 25, 1643 (1983)

    Article  MATH  Google Scholar 

  38. Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009)

    Article  MATH  Google Scholar 

  39. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)

    Article  Google Scholar 

  40. Montina, A., Bortolozzo, U., Residori, S., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    Article  MathSciNet  Google Scholar 

  41. Zakharov, V.E.: Exact solutions in the problem of parametric interaction of three-dimensional wave packets. Dokl. Acad. Nauk SSSR 228(6), 1314–1316 (1976)

    MathSciNet  Google Scholar 

  42. Craik, A.D.D., Adam, J.A.: Evolution in space and time of resonant wave triads-I. The ’pump-wave approximation’. Proc. R. Soc. A 363, 243–255 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Let. A 379, 1975–1978 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ma, W.X., Qin, Z.Y., Lv, X.: Lump solutions to dimensionally reduced pgKP and pgbKP equations. Nonlinear Dyn. 84, 923–931 (2016)

    Article  Google Scholar 

  46. Wang, C.J.: Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation. Nonlinear Dyn. 84, 697–702 (2016)

    Article  MathSciNet  Google Scholar 

  47. Wang, C.J., Dai, Z.D., Liu, C.F.: Interaction between kink solitary wave and Rogue wave for (2+1)-dimensional Burgers equation. Mediterr. J. Math. 13, 1087–1098 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lü, J., Bilige, S., Chaolu, T.: The study of lump solution and interaction phenomenon to (2\(+\)1)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn. 91, 1669–1676 (2018)

    Article  Google Scholar 

  49. Tang, Y.N., Tao, S.Q., Guan, Q.: Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput. Math. Appl. 72, 2334–2342 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang, Y., Dong, H.H., Zhang, X.E., et al.: Rational solutions and lump solutions to the generalized (3 + 1)-dimensional Shallow water-like equation. Comput. Math. Appl. 73, 246–252 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. Huang, L.L., Chen, Y.: Lump solutions and interaction phenomenon for (2+1)-dimensional SawadaKotera equation. Commun. Theor. Phys. 67(5), 473–478 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lv, J.Q., Bilige, S.D.: Lump solutions of a (2+1)-dimensional BSK equation. Nonlinear Dyn. 90, 2119–2124 (2017)

    Article  MathSciNet  Google Scholar 

  53. Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. Lü, X., Ma, W.X., Zhou, Y., Khalique, C.M.: Rational solutions to an extended Kadomtsev–Petviashvili-like equation with symbolic computation. Comput. Math. Appl. 71, 1560–1567 (2016)

    Article  MathSciNet  Google Scholar 

  55. Lü, X., Ma, W.X., Chen, S.T., Khalique, C.M.: A note on rational solutions to a Hirota–Satsuma-like equation. Appl. Math. Let. 58, 13–18 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  56. Lin, F.H., Chen, S.T., Qu, Q.X., Wang, J.P., Zhou, X.W., Lü, X.: Resonant multiple wave solutions to a new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation: linear superposition principle. Appl. Math. Lett. 78, 112–117 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. Lü, X., Wang, J.P., Lin, F.H., Zhou, X.W.: Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water. Nonlinear Dyn. 91, 1249–1259 (2018)

    Article  Google Scholar 

  58. Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili-Boussinesq equation. Nonlinear Dyn. 86, 523–534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  59. Dai, Z.D., Liu, J., Zeng, X.P., Liu, Z.J.: Periodic kink-wave and kinky periodic-wave solutions for the Jimbo–Miwa equation. Phys. Lett. A 372, 5984–5986 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  60. Wang, C.J., Dai, Z.D., Mu, G., Lin, S.-Q.: New exact periodic solitary-wave solutions for new (2+1)-dimensional KdV equation. Commun. Theor. Phys. 52, 862–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, PO BOX 19395-3697, Tehran, Iran

    Mohammadreza Foroutan

  2. Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

    Jalil Manafian

  3. Department of Mechanical Engineering, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

    Arash Ranjbaran

Authors
  1. Mohammadreza Foroutan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jalil Manafian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arash Ranjbaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Manafian.

Ethics declarations

Conflict of interests

The authors declare that there are no conflict of interests regarding the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Foroutan, M., Manafian, J. & Ranjbaran, A. Lump solution and its interaction to (3+1)-D potential-YTSF equation. Nonlinear Dyn 92, 2077–2092 (2018). https://doi.org/10.1007/s11071-018-4182-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4182-5

Keywords

Search

Navigation