Skip to main content
SpringerLink
Log in

A novel (2+1)-dimensional Sawada-Kotera type system: multisoliton solution and variable separation solution

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

Abstract

A novel (2+1)-dimensional system of the Sawada-Kotera type is considered. The existence of three-soliton and four-soliton solutions with wave number constraints is confirmed. Other interesting solutions, such as the long-range interaction between a line soliton and a y-periodic soliton, are also presented based on the Hirota formalism. By extending the multilinear variable separation approach to the fifth-order nonlinear evolution equation, various localized excitations are introduced, including solitoff, dromion, and an instanton excited by three resonant dromions. In addition to these localized excitations, the general fusion or fission type N-solitary wave solution is obtained, the Y-shaped resonant soliton and the T-type resonant soliton interaction in shallow water are graphically explored.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data Availibility

Our manuscript has no associated data.

References

  1. Sawada, K., Kotera, T.: A method for finding \(N\)-soliton solutions of the K.d.V. equation and K.d.V.-like equation. Prog. Theoret. Phys. 51, 1355–1367 (1974)

    MathSciNet  Google Scholar 

  2. Caudrey, P.J., Dodd, R.K., Gibbon, J.D.: A new hierarchy of Korteweg-de Vries equations. Proc. R. Soc. London A 351, 407–422 (1976)

    MathSciNet  Google Scholar 

  3. Lou, S.Y.: Abundant symmetries for the (1+1)-Dimensional classical Liouville field theory. J. Math. Phys. 35, 2336–2348 (1994)

    MathSciNet  Google Scholar 

  4. Grimshaw, R., Pelinovsky, E., Poloukhina, O.: Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface. Nonlinear Process. Geophys. 9, 221–235 (2002)

    Google Scholar 

  5. Satsuma, J., Kaup, D.J.: A Bäcklund transformation for a higher order Korteweg-de Vries equation. J. Phys. Soc. Jpn. 43, 692–697 (1977)

    Google Scholar 

  6. Fuchssteiner, B., Oevel, W.: The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants. J. Math. Phys. 23, 358–363 (1982)

    MathSciNet  Google Scholar 

  7. Tian, K., Liu, Q.P.: A supersymmetric Sawada-Kotera equation. Phys. Lett. A 373, 1807–1810 (2009)

    MathSciNet  Google Scholar 

  8. Wang, W., Yao, R.X., Lou, S.Y.: Abundant traveling wave structures of (1+1)-dimensional Sawada-Kotera equation: few cycle solitons and soliton molecules. Chin. Phys. Lett. 37, 100501 (2020)

    Google Scholar 

  9. Lou, S.Y.: Soliton molecules and asymmetric solitons in three fifth order systems via velocity resonance. J. Phys. Commun. 4, 041002 (2020)

    Google Scholar 

  10. Konopelehenko, B.G., Dubrovsky, V.G.: Some new integrable nonlinear evolution equations in 2+1-dimensions. Phys. Lett. A 102, 15–17 (1984)

    MathSciNet  Google Scholar 

  11. Geng, X.G.: Darboux transformation of the two-dimensional Sawada-Kotera equation. Appl. Math. J. Chin. Univ. 4, 494–497 (1989)

    Google Scholar 

  12. Nucci, M.C.: Painlevé property and pseudopotentials for nonlinear evolution equations. J. Phys. A 22, 2897 (1989)

    MathSciNet  Google Scholar 

  13. Yao, R.X., Li, Y., Lou, S.Y.: A new set and new relations of multiple soliton solutions of (2+1)-dimensional Sawada-Kotera equation. Commun. Nonlinear Sci. 99, 105820 (2021)

    MathSciNet  Google Scholar 

  14. Qi, Z., Chen, Q., Wang, M., Li, B.: New mixed solutions generated by velocity resonance in the (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 108, 1617–1626 (2022)

    Google Scholar 

  15. Yin, Z., Tian, S.F.: Nonlinear wave transitions and their mechanisms of (2+1)-dimensional Sawada-Kotera equation. Physica D 427, 133002 (2021)

    MathSciNet  Google Scholar 

  16. Lou, S.Y.: On the coherent structures of the Nizhnik-Novikov-Veselov equation. Phys. Lett. A 277, 94–100 (2000)

    MathSciNet  Google Scholar 

  17. Lou, S.Y., Ruan, H.Y.: Revisitation of the localized excitations of the (2+1)-dimensional KdV equation. J. Phys. A 34, 305 (2000)

    MathSciNet  Google Scholar 

  18. Lou, S.Y.: Dromions, dromion lattice, breathers and instantons of the Davey-Stewartson equation. Phys. Scr. 65, 7 (2000)

    MathSciNet  Google Scholar 

  19. Tang, X.Y., Lou, S.Y., Zhang, Y.: Localized excitations in (2+1)-dimensional systems. Phys. Rev. E 66, 046601 (2002)

    MathSciNet  Google Scholar 

  20. Tang, X.Y., Lou, S.Y.: Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems. J. Math. Phys. 44, 4000–4025 (2003)

    MathSciNet  Google Scholar 

  21. Cui, C.J., Tang, X.Y., Cui, Y.J.: New variable separation solutions and wave interactions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Appl. Math. Lett. 102, 106109 (2020)

    MathSciNet  Google Scholar 

  22. Tang, X.Y., Li, J., Liang, Z.F., Wang, J.Y.: Variable separation solutions and interacting waves of a coupled system of the modified KdV and potential BLMP equations. Phys. Lett. A 378, 1439–1447 (2014)

    MathSciNet  Google Scholar 

  23. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Google Scholar 

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

    Google Scholar 

  25. Hietarinta, J.: Introduction to the Hirota bilinear method: integrability of nonlinear system. Lecture Notes in Physics, Springer-Verlag, Berlin 495, 95–103 (1997)

    Google Scholar 

  26. Li, J.H., Lou, S.Y.: Kac-Moody-Virasoro symmetry algebra of a (2+1)-dimensional bilinear system. Chin. Phys. B 17, 747–753 (2008)

    Google Scholar 

  27. Sarma, D.: Constructing a supersymmetric integrable system from the Hirota method in superspace. Nucl. Phys. B 681, 351–358 (2004)

    MathSciNet  Google Scholar 

  28. Wang, J.Y., Tang, X.Y., Liang, Z.F.: Constructing (2+1)-dimensional supersymmetric integrable systems from the Hirota formalism in the superspace. Chin. Phys. B 27, 040203 (2018)

    Google Scholar 

  29. Wang, X.B., Zhao, Q., Jia, M., Lou, S.Y.: Novel travelling wave structures for (2+1)-dimensional Sawada-Kotera equation. Appl. Math. Lett. 124, 107638 (2022)

    MathSciNet  Google Scholar 

  30. Ruan, H.Y.: Interaction between line soliton and Y-periodic soliton: solutions to the asymmetric Nizhnik-Novikov-Veselov equation. Phys. Scr. 67, 240 (2003)

    Google Scholar 

  31. Li, W.T., Li, B.: Soliton solutions of weakly bound states for higher-order Ito equation. Nonlinear Dyn. 110, 741–751 (2022)

    Google Scholar 

  32. Sun, Y., Li, B.: Creation of anomalously interacting lumps by degeneration of lump chains in the BKP equation. Nonlinear Dyn. 111, 19297–19313 (2023)

    Google Scholar 

  33. Peng, W.Q., Tian, S.F., Zhang, T.T.: Dynamics of the soliton waves, breather waves, and rogue waves to the cylindrical Kadomtsev-Petviashvili equation in pair-ion-electron plasma. Phys. Fluids 31, 102107 (2019)

    Google Scholar 

  34. Chow, K.W.: Solitoff solutions of nonlinear evolution equations. J. Phys. Soc. Jpn. 65, 1971–1976 (1996)

    MathSciNet  Google Scholar 

  35. Wang, S., Tang, X.Y., Lou, S.Y.: Soliton fission and fusion: Burgers equation and Sharma-Tasso-Olver equation. Chaos Solitons Fract. 21, 231–239 (2004)

    MathSciNet  Google Scholar 

  36. Zhang, Z., Li, Z.Q., Li, B.: Fusion and fission phenomena for (2+1)-dimensional fifth-order KdV system. Appl. Math. Lett. 116, 107004 (2021)

    MathSciNet  Google Scholar 

  37. Li, Y.H., An, H.L., Zhang, Y.Y.: Abundant fission and fusion solutions in the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation. Nonlinear Dyn. 108, 2489–2503 (2022)

    Google Scholar 

  38. Kodama, Y.: KP solitons in shallow water. J. Phys. A 43, 434004 (2010)

    MathSciNet  Google Scholar 

  39. Sun, J.C., Tang, X.Y., Chen, Y.: Exploring two-dimensional internal waves: a new three-coupled Davey-Stewartson system and physics-informed neural networks with weight assignment methods. Physica D 459, 134021 (2023)

    MathSciNet  Google Scholar 

  40. Alonso, L.M., Reus, E.M.: Localized coherent structures of the Davey-Stewartson equation in the bilinear formalism. J. Math. Phys. 33, 2947–2957 (1992)

    MathSciNet  Google Scholar 

  41. Cheng, X.P., Lou, S.Y., Chen, C.L., Tang, X.Y.: Interactions between solitons and other nonlinear Schrödinger waves. Phys. Rev. E 89, 043202 (2014)

    Google Scholar 

  42. Lou, S.Y., Cheng, X.P., Tang, X.Y.: Dressed dark solitons of the defocusing nonlinear Schrödinger equation. Chin. Phys. Lett. 31, 070201 (2014)

    Google Scholar 

  43. Lou, S.Y.: Consistent Riccati expansion for integrable systems. Stud. Appl. Math. 134, 372–402 (2015)

    MathSciNet  Google Scholar 

  44. Chen, J.B., Pelinovsky, D.E.: Rogue periodic waves of the focusing nonlinear Schrödinger equation. Proc. R. Soc. A 474, 20170814 (2018)

    Google Scholar 

  45. Yang, Y.Q., Dong, H.H., Chen, Y.: Darboux-Bäcklund transformation and localized excitation on the periodic wave background for the nonlinear Schrödinger equation. Wave Motion 106, 102787 (2021)

    Google Scholar 

  46. Peng, W.Q., Pu, J.C., Chen, Y.: PINN deep learning method for the Chen-Lee-Liu equation: rogue wave on the periodic background. Commun. Nonlinear Sci. 105, 106067 (2022)

    MathSciNet  Google Scholar 

  47. Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)

    MathSciNet  Google Scholar 

  48. Fu, Z.T., Liu, S.K., Liu, S.D., Zhao, Q.: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 290, 72–76 (2001)

  49. Dai, C.Q., Wang, Y.Y.: Notes on the equivalence of different variable separation approaches for nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simulat. 19, 19–28 (2014)

  50. Radha, R., Singh, S., Kumar, C.S., Lou, S.Y.: Elusive exotic structures and their collisional dynamics in (2+1)-dimensional Boiti-Leon-Pempinelli equation. Phys. Scr. 97, 125211 (2022)

    Google Scholar 

  51. Zhong, W.P., Belić, M.: Nonlinear wave excitations in the (2+1)-D asymmetric Nizhnik-Novikov-Veselov system. Chaos Solitons Fract. 176, 114075 (2023)

    MathSciNet  Google Scholar 

  52. Quan, J.F., Tang, X.Y.: New variable separation solutions and localized waves for (2+1)-dimensional nonlinear systems by a full variable separation approach. Phys. Scr. 98, 125269 (2023)

    Google Scholar 

Download references

Acknowledgements

The authors would like to sincerely thank Junchao Sun for providing valuable comments.

Funding

This work is supported by the National Natural Science Foundation of China (No. 12075208, No. 12275085 and No. 12235007) and Science and Technology Commission of Shanghai Municipality (No. 21JC1402500 and No. 22DZ2229014).

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Quzhou University, Quzhou, 324000, People’s Republic of China

    Jianyong Wang & Yong Chen

  2. School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    Yunqing Yang

  3. School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China

    Xiaoyan Tang & Yong Chen

Authors
  1. Jianyong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunqing Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoyan Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Yang, Y., Tang, X. et al. A novel (2+1)-dimensional Sawada-Kotera type system: multisoliton solution and variable separation solution. Nonlinear Dyn 112, 8481–8494 (2024). https://doi.org/10.1007/s11071-024-09511-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-024-09511-0

Keywords

Search

Navigation