Skip to main content
SpringerLink
Log in

Wronskian and linear superposition solutions to generalized KP and BKP equations

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

Abstract

We aim to find exact solutions to some generalized KP- and BKP-type equations by the Wronskian technique and the linear superposition principle. By applying Wronskian identities of the bilinear KP hierarchy, three Wronskian formulations are first furnished, with all generating functions for matrix entries satisfying a system of combined linear partial differential equations. Second, we apply the linear superposition principle to exponential traveling wave solutions of the introduced generalized KP and BKP equations. Each exponential wave in the Wronskian and N-wave solutions satisfies the corresponding nonlinear dispersion relation.

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

Similar content being viewed by others

References

  1. Freeman, N.C., Nimmo, J.J.C.: Soliton solutions of the Korteweg–de Vries and Kadomtsev–Petviashvili equations: the Wronskian technique. Phys. Lett. A 95, 1–3 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nimmo, J.J.C., Freeman, N.C.: A method of obtaining the \(N\)-soliton solution of the Boussinesq equation in terms of a Wronskian. Phys. Lett. A 95, 4–6 (1983)

    Article  MathSciNet  Google Scholar 

  3. Ma, W.X., You, Y.: Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals 22, 395–406 (2004)

  4. Zhang, Y., Chu, L.J., Guo, B.L.: Positons, negatons and complexitons of the mKdV equation with non-uniformity terms. Appl. Math. Comput. 217, 1463–1469 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Ma, W.X.: Complexiton solutions to integrable equations. Nonlinear Anal. 63, e2461–e2471 (2005)

    Article  MATH  Google Scholar 

  6. Ma, W.X.: Complexiton solutions to the Korteweg-de Vries equation. Phys. Lett. A 301, 35–44 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, L., Zhang, Y.: Rational and complexiton solutions of the (3 + 1)-dimensional KP equation. Nonlinear Dyn. 71, 605–613 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nimmo, J.J.C., Zhao, J.X.: Determinant and Pfaffian solutions of soliton equations. Phys. Scr. 89, 038005 (2014)

    Article  Google Scholar 

  9. Kang, Y.L., Zhang, Y., Jin, L.G.: Soliton solution to BKP equation in Wronskian form. Appl. Math. Comput. 224, 250–258 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  11. Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950–959 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ma, W.X., Abdeljabbar, A., Asaad, M.G.: Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl. Math. Comput. 217, 10016–10023 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Ma, W.X., Xia, T.C.: Pfaffianized systems for a generalized Kadomtsev–Petviashvili equation. Phys. Scr. 87, 055003 (2013)

    Article  MATH  Google Scholar 

  14. Wazwaz, A.M., El-Tantawy, S.A.: A new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 84, 1107–1112 (2016)

  15. Wazwaz, A.M.: Multiple-soliton solutions for a (3 + 1)-dimensional generalized KP equation. Commun. Nonlinear Sci. Numer. Simul. 17, 491–495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wazwaz, A.M.: Distinct kinds of multiple-soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation. Phys. Scr. 84, 055006 (2011)

    Article  MATH  Google Scholar 

  17. Asaad, M.G., Ma, W.X.: Pfaffian solutions to a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation and its modified counterpart. Appl. Math. Comput. 218, 5524–5542 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Wazwaz, A.M.: Two forms of (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation: multiple soliton solutions. Phys. Scr. 86, 035007 (2012)

    Article  MATH  Google Scholar 

  19. Wazwaz, A.M.: Variants of a (3 + 1)-dimensional generalized BKP equation: multiple-front waves solutions. Comput. Fluids 97, 164–167 (2014)

    Article  MathSciNet  Google Scholar 

  20. Tang, Y.N., Ma, W.X., Xu, W., Gao, L.: Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo–Miwa equation. Appl. Math. Comput. 217, 8722–8730 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Tang, Y.N., Tu, J.Y., Ma, W.X.: Two new Wronskian conditions for the (3 + 1)-dimensional Jimbo–Miwa equation. Appl. Math. Comput. 218, 10050–100556 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Wu, J.P.: A new Wronskian condition for a (3 + 1)-dimensional nonlinear evolution equation. Chin. Phys. Lett. 28, 050501 (2011)

    Article  Google Scholar 

  23. Wazwaz, A.M.: Multiple-soliton solutions for extended (3 + 1)-dimensional Jimbo–Miwa equations. Appl. Math. Lett. 64, 21–26 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dorrizzi, B., Grammaticos, B., Ramani, A., Winternitz, P.: Are all the equations of the KP hierarchy integrable? J. Math. Phys. 27, 2848–2852 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lü, X., Ma, W.X., Yu, J., Khalique, C.M.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Ma, W.X., Zhang, Y., Tang, Y.N., Tu, J.Y.: Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218, 7174–7183 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Zhang, L.J., Khaliquey, C.M., Ma, W.X.: Classifying bilinear differential equations by linear superposition principle. Int. J. Mod. Phys. B 30, 1640029 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gao, L.N., Zhao, X.Y., Zi, Y.Y., Yu, J., Lü, X.: Resonant behavior of multiple wave solutions to a Hirota bilinear equation. Comput. Math. Appl. 72, 1225–1229 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, Y., Ma, W.X.: Applications of linear superposition principle to resonant solitons and complexitons. Comput. Math. Appl. 73, 1697–1706 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ma, W.X., Abdeljabbar, A.: A bilinear Bäcklund transformation of a (3 + 1)-dimensional generalized KP equation. Appl. Math. Lett. 25, 1500–1504 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lü, X.: New bilinear Bäcklund transformation with multisoliton solutions for the (2 + 1)-dimensional Sawada–Kotera model. Nonlinear Dyn. 76, 161–168 (2014)

    Article  MATH  Google Scholar 

  33. Lü, X., Lin, F.H., Qi, F.H.: Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions. Appl. Math. Model. 39, 3221–3226 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Ma, W.X., Zhou, Y., Dougherty, R.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Mod. Phys. B 30, 1640018 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gilson, C.R., Nimmo, J.J.C.: Lump solutions of the BKP equation. Phys. Lett. A 147, 472–476 (1990)

    Article  MathSciNet  Google Scholar 

  37. 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 

  38. Ma, W.X.: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2, 140–144 (2011)

    Google Scholar 

  39. Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72, 41–56 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the Referees and Editor for their valuable comments. This work is supported by the National Natural Science Foundation of China (No. 11371326).

Author information

Authors and Affiliations

  1. Normal School, Jinhua Polytechnic, Jinhua, 321007, China

    Li Cheng

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Yi Zhang

Authors
  1. Li Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Cheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, L., Zhang, Y. Wronskian and linear superposition solutions to generalized KP and BKP equations. Nonlinear Dyn 90, 355–362 (2017). https://doi.org/10.1007/s11071-017-3666-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3666-z

Keywords

Search

Navigation