Advertisement

Pramana

, 93:4 | Cite as

Pfaffians of B-type Kadomtsev–Petviashvili equation and complexitons to a class of nonlinear partial differential equations in (3\(+\)1) dimensions

  • Li ChengEmail author
  • Yi Zhang
  • Wen-Xiu Ma
Article
  • 13 Downloads

Abstract

The aim of this paper is to investigate a class of generalised Kadomtsev–Petviashvili (KP) and B-type Kadomtsev–Petviashvili (BKP) equations, which include many important nonlinear evolution equations as its special cases. By applying the fundamental Pfaffian identity, a general Pfaffian formulation is established and all the involved generating functions for Pfaffian entries need to satisfy a system of combined linear partial differential equations. The illustrative examples of the presented Pfaffian solutions are given for the (3\(+\)1)-dimensional generalised KP, Jimbo–Miwa and BKP equations. Moreover, we use the linear superposition principle to generate exponential travelling wave solutions and mixed resonant solutions of the considered equations.

Keywords

Generalised Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations Pfaffian formulation sufficient conditions N-wave solutions complexitons 

PACS Nos

02.30.Ik 05.45.Yv 

Notes

Acknowledgements

The authors express their sincere thanks to the referees and editors for their valuable comments. This work was supported by the National Natural Science Foundation of China (No. 11371326).

References

  1. 1.
    N C Freeman and J J C Nimmo, Phys. Lett. A 95, 1 (1983)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    J J C Nimmo and N C Freeman, Phys. Lett. A 95, 4 (1983)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    W X Ma, A Abdeljabbar and M G Asaad, Appl. Math. Comput. 217, 10016 (2011)MathSciNetGoogle Scholar
  4. 4.
    W X Ma and Y You, Chaos Solitons Fractals 22, 395 (2004)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    M Li, J H Xiao, W J Liu, Y Jiang and B Tian, Commun. Nonlinear Sci. Numer. Simul.  17, 2845 (2012)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    M G Asaad and W X Ma, Appl. Math. Comput. 218, 5524 (2012)MathSciNetGoogle Scholar
  7. 7.
    W X Ma and T C Xia, Phys. Scr. 87, 055003 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    L Cheng and Y Zhang, Nonlinear Dyn. 90, 355 (2017)CrossRefGoogle Scholar
  9. 9.
    M Adler, T Shiota and M P Van, Math. Ann. 322, 423 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C R Gilson and J J C Nimmo, Phys. Lett. A 147, 472 (1990)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    H Gao, Pramana – J. Phys.  88: 84 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    R Hirota, The direct method in soliton theory (Cambridge University Press, New York, 2004)CrossRefGoogle Scholar
  13. 13.
    B Dorizzi, B Grammaticos, A Ramani and P Winternitz, J. Math. Phys. 27, 2848 (1986)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    W X Ma and E G Fan, Comput. Math. Appl. 61, 950 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    B Tian, Y T Gao and W Hong, Comput. Math. Appl. 44, 525 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A M Wazwaz, Phys. Scr. 84, 055006 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Z H Xu, H L Chen and Z D Dai, Pramana – J. Phys. 87: 31 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    A M Wazwaz and S A El-Tantawy, Nonlinear Dyn. 84, 1107 (2016)CrossRefGoogle Scholar
  19. 19.
    A M Wazwaz, Appl. Math. Lett. 64, 21 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    W X Ma, Y Zhang, Y N Tang and J Y Tu, Appl. Math. Comput. 218, 7174 (2012)MathSciNetGoogle Scholar
  21. 21.
    L J Zhang, C M Khalique and W X Ma, Int. J. Mod. Phys. B  30, 1640029 (2016)ADSCrossRefGoogle Scholar
  22. 22.
    W X Ma, Nonlinear Anal. 63, e2461 (2005)CrossRefGoogle Scholar
  23. 23.
    W X Ma, Phys. Lett. A 301, 35 (2002)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Y Zhou and W X Ma, Comput. Math. Appl. 73, 1697 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    H C Zheng, W X Ma and X Gu, Appl. Math. Comput. 220, 226 (2013)MathSciNetGoogle Scholar
  26. 26.
    W X Ma and Z N Zhu, Appl. Math. Comput. 218, 11871 (2012)MathSciNetGoogle Scholar
  27. 27.
    A M Wazwaz, Phys. Scr. 86, 035007 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    A R Seadawy, Pramana – J. Phys. 89: 49 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    M S Osman, Pramana – J. Phys. 88: 67 (2017)ADSCrossRefGoogle Scholar
  30. 30.
    W X Ma and A Abdeljabbar, Appl. Math. Lett. 25, 1500 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    X Lü, B Tian and F H Qi, Nonlinear Anal. Real World Appl. 13, 1130 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    W X Ma, Phys. Lett. A 379, 1975 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    J Y Yang and W X Ma, Int. J. Mod. Phys. B 30, 1640028 (2016)ADSCrossRefGoogle Scholar
  34. 34.
    W X Ma, Y Zhou and R Dougherty, Int. J. Mod. Phys. B 30, 1640018 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    J Y Yang, W X Ma and Z Y Qin, Anal. Math. Phys. 8, 427 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    W X Ma and Y Zhou, J. Differ. Equ. 264, 2633 (2018)ADSCrossRefGoogle Scholar
  37. 37.
    W X Ma, J. Geom. Phys. 133, 10 (2018)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    W Tan, H P Dai, Z D Dai and W Y Zhong, Pramana – J. Phys. 89: 77 (2017)ADSCrossRefGoogle Scholar
  39. 39.
    W X Ma, X L Yong and H Q Zhang, Comput. Math. Appl. 75, 289 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    J Y Yang and W X Ma, Nonlinear Dyn. 89, 1539 (2017)CrossRefGoogle Scholar
  41. 41.
    W X Ma, Int. J. Nonlin. Sci. Numer. 17, 355 (2016)CrossRefGoogle Scholar
  42. 42.
    L Cheng and Y Zhang, Mod. Phys. Lett. B 31, 1750224 (2017)ADSCrossRefGoogle Scholar
  43. 43.
    X Lü, S T Chen and W X Ma, Nonlinear Dyn. 86, 523 (2016)CrossRefGoogle Scholar
  44. 44.
    X Lü and W X Ma, Nonlinear Dyn. 85, 1217 (2016)CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Normal SchoolJinhua PolytechnicJinhuaChina
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  3. 3.Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical ModellingNorth-West UniversityMmabathoSouth Africa
  4. 4.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  5. 5.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina

Personalised recommendations