Skip to main content
SpringerLink
Log in

Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

Abundant exact interaction solutions, including lump-soliton, lump-kink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2 + 1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

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. Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge Univ Press, 1991

    Book  MATH  Google Scholar 

  2. Caudrey P J. Memories of Hirota’s method: application to the reduced Maxwell-Bloch system in the early 1970s. Philos Trans Roy Soc A, 2011, 369: 1215–1227

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen S T, Ma W X. Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation. Front Math China, 2018, 13: 525–534

    Article  MathSciNet  MATH  Google Scholar 

  4. Dong H H, Zhang Y, Zhang X E. The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation. Commun Nonlinear Sci Numer Simul, 2016, 36: 354–365

    Article  MathSciNet  Google Scholar 

  5. Dorizzi B, Grammaticos B, Ramani A, Winternitz P. Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? J Math Phys, 1986, 27: 2848–2852

    MathSciNet  MATH  Google Scholar 

  6. Drazin P G, Johnson R S. Solitons: An Introduction. Cambridge: Cambridge Univ Press, 1989

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Harun-Or-Roshid, Ali M Z. Lump solutions to a Jimbo-Miwa like equation. arXiv:1611.04478

  9. Hietarinta J. Introduction to the Hirota bilinear method. In:Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani K M, eds. Integrability of Nonlinear Systems. Lecture Notes in Phys, Vol 495. Berlin: Springer, 1997, 95–103

    Chapter  Google Scholar 

  10. Hirota R. The Direct Method in Soliton Theory. New York: Cambridge Univ Press. 2004

    Book  MATH  Google Scholar 

  11. Ibragimov N H. A new conservation theorem. J Math Anal Appl, 2007, 333: 311–328

    Article  MathSciNet  MATH  Google Scholar 

  12. Imai K. Dromion and lump solutions of the Ishimori-I equation. Prog Theor Phys, 1997, 98: 1013–1023

    Article  Google Scholar 

  13. Kaup D J. The lump solutions and the Backlund transformation for the three-dimensional three-wave resonant interaction. J Math Phys, 1981, 22: 1176–1181

    Article  MathSciNet  MATH  Google Scholar 

  14. Kofane T C, Fokou M, Mohamadou A, Yomba E. Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation. Eur Phys J Plus, 2017, 132: 465

    Article  Google Scholar 

  15. Konopelchenko B, Strampp W. The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Problems, 1991, 7: L17–L24

    Article  MathSciNet  MATH  Google Scholar 

  16. Li X Y, Zhao Q L. A new integrable symplectic map by the binary nonlinearization to the super AKNS system. J Geom Phys, 2017, 121: 123–137

    Article  MathSciNet  MATH  Google Scholar 

  17. Li X Y, Zhao Q L, Li Y X, Dong H H. Binary Bargmann symmetry constraint associated with 3 × 3 discrete matrix spectral problem. J Nonlinear Sci Appl, 2015, 8: 496–506

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu J G, Zhou L, He Y. Multiple soliton solutions for the new (2 + 1)-dimensional Korteweg-de Vries equation by multiple exp-function method. Appl Math Lett, 2018, 80: 71–78

    Article  MathSciNet  MATH  Google Scholar 

  19. Lu X, Chen S T, Ma W X. Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dynam, 2016, 86: 523–534

    Article  MathSciNet  MATH  Google Scholar 

  20. 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, 2016, 58: 13–18

    Article  MathSciNet  MATH  Google Scholar 

  21. 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, 2016, 71: 1560–1567

    Article  MathSciNet  Google Scholar 

  22. Ma W X. Wronskian solutions to integrable equations. Discrete Contin Dyn Syst, 2009, Suppl: 506–515

  23. Ma W X. Conservation laws of discrete evolution equations by symmetries and adjoint symmetries. Symmetry, 2015, 7: 714–725

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Ma W X. Lump-type solutions to the (3 + 1)-dimensional Jimbo-Miwa equation. Int J Nonlinear Sci Numer Simul, 2016, 17: 355–359

    Article  MathSciNet  MATH  Google Scholar 

  26. Ma W X. Conservation laws by symmetries and adjoint symmetries. Discrete Contin Dyn Syst Ser S, 2018, 11: 707–721

    MathSciNet  MATH  Google Scholar 

  27. Ma W X. Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system. J Geom Phys, 2018, 132: 45–54

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma W X. Abundant lumps and their interaction solutions of (3 + 1)-dimensional linear PDEs. J Geom Phys, 2018, 133: 10–16

    Article  MathSciNet  MATH  Google Scholar 

  29. Ma W X. Diverse lump and interaction solutions to linear PDEs in (3 + 1)-dimensions. East Asian J Appl Math, 2019, 9: 185–194

    Article  MathSciNet  Google Scholar 

  30. Ma W X. Lump and interaction solutions to linear (4 + 1)-dimensional PDEs. Acta Math Sci Ser B Engl Ed, 2019, 39: 498–508

    Google Scholar 

  31. Ma W X. A search for lump solutions to a combined fourth-order nonlinear PDE in (2 + 1)-dimensions. J Appl Anal Comput, 2019, 9(to appear)

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Ma W X, Li J, Khalique C M. A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in (2 + 1)-dimensions. Complexity, 2018, 2018: 9059858

    MATH  Google Scholar 

  34. Ma W X, Qin Z Y, Lu X. Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dynam, 2016, 84: 923–931

    Article  MathSciNet  MATH  Google Scholar 

  35. Ma W X, Yong X L, Zhang H Q. Diversity of interaction solutions to the (2 + 1)-dimensional Ito equation. Comput Math Appl, 2018, 75: 289–295

    Article  MathSciNet  Google Scholar 

  36. Ma W X, You Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans Amer Math Soc, 2005, 357: 1753–1778

    Article  MathSciNet  MATH  Google Scholar 

  37. Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264: 2633–2659

    Article  MathSciNet  MATH  Google Scholar 

  38. Ma W X, Zhou Y, Dougherty R. Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Internat J Modern Phy B, 2016, 30: 1640018

    Article  MathSciNet  MATH  Google Scholar 

  39. Manakov S V, Zakharov V E, Bordag L A, Matveev V B. Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys Lett A, 1977, 63: 205–206

    Article  Google Scholar 

  40. Manukure S, Zhou Y, Ma W X. Lump solutions to a (2 + 1)-dimensional extended KP equation. Comput Math Appl, 2018, 75: 2414–2419

    Article  MathSciNet  MATH  Google Scholar 

  41. Novikov S, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons—The Inverse Scattering Method. New York: Consultants Bureau, 1984

    MATH  Google Scholar 

  42. Satsuma J, Ablowitz M J. Two-dimensional lumps in nonlinear dispersive systems. J Math Phys, 1979, 20: 1496–1503

    Article  MathSciNet  MATH  Google Scholar 

  43. Tan W, Dai H P, Dai Z D, Zhong W Y. Emergence and space-time structure of lump solution to the (2 + 1)-dimensional generalized KP equation. Pramana—J Phys, 2017, 89: 77

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Ünsal Ö, Ma W X. Linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations. Comput Math Appl, 2016, 71: 1242–1247

    Article  MathSciNet  Google Scholar 

  46. Wang D S, Yin Y B. Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach. Comput Math Appl, 2016, 71: 748–757

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu Z H, Chen H L, Dai Z D. Rogue wave for the (2 + 1)-dimensional Kadomtsev-Petviashvili equation. Appl Math Lett, 2014, 37: 34–38

    Article  MathSciNet  MATH  Google Scholar 

  49. Yang J Y, Ma W X. Lump solutions of the BKP equation by symbolic computation. Internat J Modern Phys B, 2016, 30: 1640028

    Article  MathSciNet  MATH  Google Scholar 

  50. Yang J Y, Ma W X. Abundant interaction solutions of the KP equation. Nonlinear Dynam, 2017, 89: 1539–1544

    Article  MathSciNet  Google Scholar 

  51. Yang J Y, Ma W X. Abundant lump-type solutions of the Jimbo-Miwa equation in (3 + 1)-dimensions. Comput Math Appl, 2017, 73: 220–225

    Article  MathSciNet  MATH  Google Scholar 

  52. Yang J Y, Ma W X, Qin Z Y. Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal Math Phys, 2018, 8: 427–436

    Article  MathSciNet  MATH  Google Scholar 

  53. Yang J Y, Ma W X, Qin Z Y. Abundant mixed lump-soliton solutions to the BKP equation. East Asian J Appl Math, 2018, 8: 224–232

    Article  MathSciNet  Google Scholar 

  54. Yong X L, Ma W X, Huang Y H, Liu Y. Lump solutions to the Kadomtsev-Petviashvili I equation with a self-consistent source. Comput Math Appl, 2018, 75: 3414–3419

    Article  MathSciNet  MATH  Google Scholar 

  55. Yu J P, Sun Y L. Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dynam, 2017, 87: 2755–2763

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang Y, Sun S L, Dong H H. Hybrid solutions of (3 + 1)-dimensional Jimbo-Miwa equation. Math Probl Eng, 2017, 2017: 5453941

    MathSciNet  Google Scholar 

  58. Zhang H Q, Ma W X. Lump solutions to the (2 + 1)-dimensional Sawada-Kotera equation. Nonlinear Dynam, 2017, 87: 2305–2310

    Article  MathSciNet  Google Scholar 

  59. Zhang J B, Ma W X. Mixed lump-kink solutions to the BKP equation. Comput Math Appl, 2017, 74: 591–596

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhao H Q, Ma W X. Mixed lump-kink solutions to the KP equation. Comput Math Appl, 2017, 74: 1399–1405

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhao Q L, Li X Y. A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal Math Phys, 2016, 6: 237–254

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11301454, 11301331, 11371086, 11571079, 51771083), NSF under the grant DMS-1664561, the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17KJB110020), Emphasis Foundation of Special Science Research on Subject Frontiers of CUMT under Grant No. 2017XKZD11, and the Distinguished Professorships by Shanghai University of Electric Power, China and North-West University, South Africa.

Author information

Authors and Affiliations

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

    Wen-Xiu Ma

  2. Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Wen-Xiu Ma

  3. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620, USA

    Wen-Xiu Ma

  4. College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Wen-Xiu Ma

  5. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Wen-Xiu Ma

  6. Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho, 2735, South Africa

    Wen-Xiu Ma

Authors
  1. Wen-Xiu Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen-Xiu Ma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, WX. Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions. Front. Math. China 14, 619–629 (2019). https://doi.org/10.1007/s11464-019-0771-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-019-0771-y

Keywords

MSC

Search

Navigation