General lump solutions, lumpoff solutions, and rogue wave solutions with predictability for the (2+1)-dimensional Korteweg-de Vries equation

  • Hui Wang
  • Shou-Fu TianEmail author
  • Tian-Tian ZhangEmail author
  • Yi Chen
  • Yong FangEmail author


In this work, the (2+1)-dimensional Korteweg-de Vries equation is investigated, which can be used to represent the amplitude of the shallow–water waves in fluids or electrostatic wave potential in plasmas. By employing the properties of Bell’s polynomial, we obtain bilinear representation of the equation with the aid of an appropriate transformation. Based on the obtained Hirota bilinear form, its lump solutions with localized characteristics are constructed in detail. We then derive the lumpoff solutions of the equation by studying a soliton solution generated by lump solutions. Furthermore, special rogue wave solutions with predictability are well presented, and the time and place of appearance are also derived. Finally, some graphic analysis is represented to better understand the propagation characteristics of the obtained solutions. It is hoped that our results provided in this work can be used to enrich the dynamic behaviors of the equation.


The (2+1)-dimensional Korteweg-de Vries equation Bilinear form Lump solutions Lumpoff solutions Rogue wave solutions 

Mathematics Subject Classification

35Q51 35Q53 35C99 68W30 74J35 



We express our sincere thanks to the Editor and Reviewers for their valuable comments. This work was supported by the Postgraduate Research and Practice of Educational Reform for Graduate students in CUMT under Grant No. 2019YJSJG046, the Natural Science Foundation of Jiangsu Province under Grant No. BK20181351, the Six Talent Peaks Project in Jiangsu Province under Grant No. JY-059, the Qinglan Project of Jiangsu Province of China, the National Natural Science Foundation of China under Grant Nos. 11975306 and 61877053, the Fundamental Research Fund for the Central Universities under the Grant Nos. 2019ZDPY07 and 2019QNA35, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and 2017T100413.

Compliance with ethical standards

Disclosure statement

No potential conflict of interest was reported by the authors.


  1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  2. Bluman GW, Kumei S (1989) Symmetries and differential equations. Springer, New YorkzbMATHCrossRefGoogle Scholar
  3. Boiti M, Leon JP, Manna M, Pempinelli F (1986) On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl 2(3):271–279MathSciNetzbMATHCrossRefGoogle Scholar
  4. Christou K, Christou MA (2017) 2D solitons in Boussinesq equation with dissipation. Comput Appl Math 36:513–523MathSciNetzbMATHCrossRefGoogle Scholar
  5. Das A (2018) Explicit Weierstrass traveling wave solutions and bifurcation analysis for dissipative Zakharov-Kuznetsov modified equal width equation. Comput Appl Math 37(3):3208–3225MathSciNetzbMATHCrossRefGoogle Scholar
  6. Dong MJ, Tian SF, Wang XB, Zhang TT (2018) Lump-type solutions and interaction solutions in the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation. Math Phys Anal.
  7. Dong MJ, Tian SF, Yan XW, Zou L (2018) Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation. Comput Math Appl 75(3):957–964MathSciNetzbMATHCrossRefGoogle Scholar
  8. Dong MJ, Tian SF, Yan XW, Zhang TT (2018) Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq-Burgers equation. Nonlinear Dyn 95(1):273–291CrossRefGoogle Scholar
  9. Dorizzi B, Grammaticos B, Ramani A, Winternitz P (1986) Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? J Math Phys 27(12):2848–2852MathSciNetzbMATHCrossRefGoogle Scholar
  10. Feng LL, Zhang TT (2018) Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation. Appl Math Lett 78:133–140MathSciNetzbMATHCrossRefGoogle Scholar
  11. Hietarinta J (1997) Introduction to the Hirota bilinear method. Springer, BerlinzbMATHCrossRefGoogle Scholar
  12. Hirota R (1971) Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192zbMATHCrossRefGoogle Scholar
  13. Hirota R (2004) The direct method in soliton theory. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  14. Jia M, Lou SY (2018) Lump, Lumpoff and Predictable Instanon/Rogue wave Solutions to KP equation. preprint, arXiv:1803.01730v1 [nlin.SI]
  15. Kaup DJ (1981) The lump solutions and the bäcklund transformation for the three-dimensional three-wave resonant interaction. J Math Phys 22(6):1176–1181MathSciNetzbMATHCrossRefGoogle Scholar
  16. Liu J, Mu G, Dai ZD, Luo HY (2016) Spatiotemporal deformation of multi-soliton to (2+1)-dimensional KdV equation. Nonlinear Dyn 83(1–2):355–360MathSciNetCrossRefGoogle Scholar
  17. Lou SY, Hu XB (1997) Infinitely many Lax pairs and symmetry constraints of the KP equation. J Math Phys 38(12):6401–6427MathSciNetzbMATHCrossRefGoogle Scholar
  18. Lü X, Ma WX (2016) Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn 85(2):1217–1222MathSciNetzbMATHCrossRefGoogle Scholar
  19. Ma WX (2015) Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A 379(36):1975–1978MathSciNetzbMATHCrossRefGoogle Scholar
  20. Ma WX, Zhou Y (2018) Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differ Equ 264(4):2633–2659MathSciNetzbMATHCrossRefGoogle Scholar
  21. Ma WX, Zhou Y, Dougherty R (2016) Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int J Mod Phys B 30:1640018MathSciNetzbMATHCrossRefGoogle Scholar
  22. Ma WX, Qin ZY, Lü X (2016) Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn 84(2):923–931MathSciNetzbMATHCrossRefGoogle Scholar
  23. Matveev VB, Salle MA (1991) Darboux transformations and solitons. Springer, BerlinzbMATHCrossRefGoogle Scholar
  24. Osman MS (2016) Multi-soliton rational solutions for some nonlinear evolution equations. Open Phys 14(1):26–36CrossRefGoogle Scholar
  25. Osman MS (2017) Analytical study of rational and double-soliton rational solutions governed by the KdV-Sawada-Kotera-Ramani equation with variable coefficients. Nonlinear Dyn 89(3):2283–2289MathSciNetCrossRefGoogle Scholar
  26. Osman MS, Machado JAT (2018) New nonautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation. Nonlinear Dyn 93(2):733–740zbMATHCrossRefGoogle Scholar
  27. Osman MS, Machado JAT (2018) The dynamical behavior of mixed-type soliton solutions described by (2+1)-dimensional Bogoyavlensky-Konopelchenko equation with variable coefficients. J Electromagn Waves Appl 32(11):1457–1464CrossRefGoogle Scholar
  28. Osman MS, Wazwaz AM (2018) An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients. Appl Math Comput 321:282–289MathSciNetzbMATHGoogle Scholar
  29. Peng WQ, Tian SF, Zhang TT (2018) Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation. EPL (Europhys. Lett.) 123(5):50005MathSciNetCrossRefGoogle Scholar
  30. Peng WQ, Tian SF, Zou L, Zhang TT (2018) Characteristics of the solitary waves and lump waves with interaction phenomena in a (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation. Nonlinear Dyn 93(4):1841–1851zbMATHCrossRefGoogle Scholar
  31. Peng WQ, Tian SF, Zhang TT (2018) Analysis on lump, lumpoff and rogue waves with predictability to the(2+1)-dimensional B-type Kadomtsev-Petviashvili equation. Phys Lett A 382(38):2701–2708MathSciNetzbMATHCrossRefGoogle Scholar
  32. Qin CY, Tian SF, Zhang TT (2018) Rogue waves, bright-dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Comput Math Appl 75(12):4221–4231MathSciNetzbMATHCrossRefGoogle Scholar
  33. Qin CY, Tian SF, Zou L, Zhang TT (2018) Lie symmetry analysis, conservation laws and exact solutions of fourth-order time fractional Burgers equation. J Appl Anal Comput 8(6):1727–1746MathSciNetGoogle Scholar
  34. Qin CY, Tian SF, Zou L, Ma WX (2018) Solitary wave and quasi-periodic wave solutions to a (3+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation. Adv Appl Math Mech 10(4):948–977MathSciNetCrossRefGoogle Scholar
  35. Tamizhmani KM, Punithavathi P (1990) The infinite-dimensional lie algebraic structure and the symmetry reduction of a nonlinear higher-dimensional equation. J Phys Soc Jpn 59:843–847MathSciNetCrossRefGoogle Scholar
  36. Tian SF (2017) Initial-boundary value problems for the general coupled nonlinear Schrödinger equationson the interval via the Fokas method. J Differ Equ 262:506–558zbMATHCrossRefGoogle Scholar
  37. Tian SF (2018) Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Commun Pure Appl Anal 17(3):923–957MathSciNetzbMATHCrossRefGoogle Scholar
  38. Tian SF (2018) Asymptotic behavior of a weakly dissipative modified two-component Dullin-Gottwald-Holm system. Appl Math Lett 83:65–72MathSciNetzbMATHCrossRefGoogle Scholar
  39. Tian SF (2019) Infinite propagation speed of a weakly dissipative modified two-component Dullin-Gottwald-Holm system. Appl Math Lett 89:1–7MathSciNetzbMATHCrossRefGoogle Scholar
  40. Tian SF, Zhang HQ (2010) Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations. J Math Anal Appl 371:585–608MathSciNetzbMATHCrossRefGoogle Scholar
  41. Tian SF, Zhang HQ (2012) On the integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation. J Phys A Math Theor 45:055203MathSciNetzbMATHCrossRefGoogle Scholar
  42. Tian SF, Zhang HQ (2014) On the integrability of a generalized variable-coefficient forced Korteweg-de Vries equation in fluids. Stud Appl Math 132:212–246MathSciNetzbMATHCrossRefGoogle Scholar
  43. Tian SF, Zhang TT (2018) Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc Am Math Soc 146(4):1713–1729zbMATHCrossRefGoogle Scholar
  44. Tu JM, Tian SF, Xu MJ, Ma PL, Zhang TT (2016a) On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev- Petviashvili equation in fluid dynamics. Comput Math Appl 72:2486–2504MathSciNetzbMATHCrossRefGoogle Scholar
  45. Tu JM, Tian SF, Xu MJ, Zhang TT (2016b) Quasi-periodic waves and solitary waves to a generalized KdV-Caudrey-Dodd-Gibbon equation from fluid dynamics. Taiwanese J Math 20:823–848MathSciNetzbMATHCrossRefGoogle Scholar
  46. Wang XB, Tian SF (2018) Lie symmetry analysis, conservation laws and analytical solutions of the time-fractional thin-film equation. Comput Appl Math 37(5):6270–6282MathSciNetzbMATHCrossRefGoogle Scholar
  47. Wang XB, Tian SF, Qin CY (2003) Zhang TT (2016) Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation. EPL 114:2Google Scholar
  48. Wang XB, Tian SF, Xu MJ, Zhang TT (2016) On integrability and quasi-periodic wave solutions to a (3+1)-dimensional generalized KdV-like model equation. Appl Math Comput 283:216–233MathSciNetzbMATHGoogle Scholar
  49. Wang XB, Tian SF, Qin CY, Zhang TT (2017) Characteristics of the breathers, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation. EPL 115(1):10002CrossRefGoogle Scholar
  50. Wang XB, Tian SF, Qin CY, Zhang TT (2017) Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation. Appl Math Lett 72:58–64MathSciNetzbMATHCrossRefGoogle Scholar
  51. Wang XB, Zhang TT, Dong MJ (2018) Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation. Appl Math Lett 86:298–304MathSciNetzbMATHCrossRefGoogle Scholar
  52. Wang XB, Tian SF, Zhang TT (2018) Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation. Proc Am Math Soc 146(8):3353–3365zbMATHCrossRefGoogle Scholar
  53. Wazwaz AM, Osman MS (2018) Analyzing the combined multi-waves polynomial solutions in a two-layer-liquid medium. Comput Math Appl 76(2):276–283MathSciNetzbMATHCrossRefGoogle Scholar
  54. Xu MJ, Tian SF, Tu JM, Zhang TT (2016) Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation. Nonlinear Anal Real World Appl 31:388–408MathSciNetzbMATHCrossRefGoogle Scholar
  55. Yan XW, Tian SF, Wang XB, Zhang TT (2018) Solitons to rogue waves transition, lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid dynamics. J Comput Math Int.
  56. Yan XW, Tian SF, Dong MJ, Zhang TT (2018) Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput Math Appl 76(1):179–186MathSciNetzbMATHCrossRefGoogle Scholar
  57. Yan XW, Tian SF, Dong MJ, Zou L (2018) Bäcklund transformation, rogue wave solutions and interaction phenomena for a (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dyn 92(2):709–720zbMATHCrossRefGoogle Scholar
  58. Yan XW, Tian SF, Dong MJ, Wang XB, Zhang TT (2018) Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin-Bona-Mahony equation. Z Naturforsch A 73(5):399–405CrossRefGoogle Scholar
  59. Yang JY, Ma WX (2017) Abundant lump-type solutions of the Jimbo-Miwa equation in (3+1)-dimensions. Comput Math Appl 73(2):220–225MathSciNetzbMATHCrossRefGoogle Scholar
  60. Yang JY, Ma WX (2017) Abundant interaction solutions of the KP equation. Nonlinear Dyn 89(2):1539–1544MathSciNetCrossRefGoogle Scholar
  61. Yang JY, Ma WX, Qin ZY (2018) Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal Math Phys 8(3):427–436MathSciNetzbMATHCrossRefGoogle Scholar
  62. Zhang JF (2000) Abundant dromion-like structures to the (2+1)-dimensional KdV equation. Chin Phys 9:1–4CrossRefGoogle Scholar
  63. Zhang Y, Liu YP, Tang XY (2018) M-lump and interactive solutions to a (3+1)-dimensional nonlinear system. Nonlinear Dyn 93(4):2533–2541zbMATHCrossRefGoogle Scholar
  64. Zhao HQ, Ma WX (2017) Mixed lump-kink solutions to the KP equation. Comput Math Appl 74(6):1399–1405MathSciNetzbMATHCrossRefGoogle Scholar
  65. Zhao ZL, Chen Y, Han B (2017) Lump soliton, mixed lump stripe and periodic lump solutions of a (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. Mod Phys Lett B 31:1750157MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematics and Institute of Mathematical PhysicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina

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