Skip to main content
SpringerLink
Log in

Hybrid localized wave solutions for a generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system in a fluid or plasma

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

Abstract

Based on the long wave limit method and complex conjugate condition technique, we investigate hybrid localized wave solutions with different forms for the generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system. Four kinds of bilinear auto-Bäcklund transformations are constructed by constructing different equivalent exchange formulas. The system simulates the formation of localized waves on the ocean surface and the interaction among water waves. In order to better analyze the dynamic characteristics of hybrid localized wave solutions, several three-dimensional diagrams are drawn with the help of Mathematica software. Besides, seven kinds of combined waves are summarized, including the hybrid solutions consisting of L-order kink waves, Q-order breather waves and M-order lump waves. Water wave phenomena can be simulated by nonlinear evolution equations. Analyzing the images of analytic solutions is helpful to understand the dynamic behavior of these models. We hope that bilinear auto-Bäcklund transformations and hybrid localized wave solutions can help researchers simulate nonlinear phenomena in the fields of hydrodynamics,oceanography,ionospheric physics, optics, condensed state physics and so on.

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

Similar content being viewed by others

Availability of data and materials

All data generated or analyzed during this study are included in this published article.

References

  1. Lü, X.: Madelung fluid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn. 81, 239–247 (2015)

    Article  MATH  Google Scholar 

  2. Liu, J.G., Zhu, W.H.: Breather wave solutions for the generalized shallow water wave equation with variable coefficients in the atmosphere, rivers, lakes and oceans. Comput. Math. Appl. 78, 848–856 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yadav, O.P., Jiwari, R.: Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation. Nonlinear Dyn. 95, 2825–2836 (2019)

    Article  MATH  Google Scholar 

  4. Zhang, R.F., Bilige, S.D.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  MATH  Google Scholar 

  5. Wang, X., Wei, J.: Antidark solitons and soliton molecules in a \((3+1)\)-dimensional nonlinear evolution equation. Nonlinear Dyn. 102, 363–377 (2020)

    Article  Google Scholar 

  6. Lü, X., Ma, W.X., Yu, J., Lin, F.H.: Khalique CM. Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82(3), 1211-1220 (2015)

  7. Kumar, S., Jiwari, R., Mittal, R.C., Awrejcewicz, J.: Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model. Nonlinear Dyn. 104, 661–682 (2021)

    Article  Google Scholar 

  8. Han, P.F., Bao, T.: Novel hybrid-type solutions for the \((3+1)\)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients. Nonlinear Dyn. 107, 1163–1177 (2022)

    Article  Google Scholar 

  9. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the \((2+1)\)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Liu, J.G., Zhu, W.H., Zhou, L.: Breather wave solutions for the Kadomtsev–Petviashvili equation with variable coefficients in a fluid based on the variable-coefficient three-wave approach. Math. Methods Appl. Sci. 43, 458–465 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Osman, M.S., Machado, J.A.T.: New nonautonomous combined multi-wave solutions for \((2+1)\)-dimensional variable-coefficients KdV equation. Nonlinear Dyn. 93(2), 733–740 (2018)

    Article  MATH  Google Scholar 

  12. Han, P.F., Bao, T.: Dynamic analysis of hybrid solutions for the new \((3+1)\)-dimensional Boiti–Leon–Manna–Pempinelli equation with time-dependent coefficients in incompressible fluid. Eur. Phys. J. Plus 136, 925 (2021)

    Article  Google Scholar 

  13. Jiwari, R., Kumar, V., Karan, R., Alshomrani, A.S.: Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie group method. Int. J. Numer. Meth. Heat Fluid Flow 27(6), 1332–1350 (2017)

    Article  Google Scholar 

  14. Zhao, X., Tian, B., Du, X.X., Hu, C.C., Liu, S.H.: Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized \((2+1)\)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Eur. Phys. J. Plus 136, 159 (2021)

    Article  Google Scholar 

  15. Wazwaz, A.M.: The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions. Appl. Math. Comput. 170, 347–360 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Han, P.F., Bao, T.: Construction of abundant solutions for two kinds of \((3+1)\)-dimensional equations with time-dependent coefficients. Nonlinear Dyn. 103, 1817–1829 (2021)

    Article  Google Scholar 

  17. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the \((3+1)\)-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

  18. Kumar, V., Gupta, R.K., Jiwari, R.: Painlevé analysis, lie symmetries and exact solutions for variable coefficients Benjamin–Bona–Mahony–Burger (BBMB) equation. Commun. Theor. Phys. 60, 175–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Verma, A., Jiwari, R., Koksal, M.E.: Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions. Adv. Differ. Equ. (2014) (in press). https://doi.org/10.1186/1687-1847-2014-229

  20. Pandit, S., Kumar, M., Mohapatra, R.N., Alshomrani, A.S. Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets. Heat Fluid Flow 27(8), 1814-1850 (2017)

    Article  Google Scholar 

  21. Han, P.F., Taogetusang: Lump-type, breather and interaction solutions to the \((3+1)\)-dimensional generalized KdV-type equation. Mod. Phys. Lett. B 34(29), 2050329 (2020)

    Article  MathSciNet  Google Scholar 

  22. Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010)

    Article  MATH  Google Scholar 

  23. Ma, W.X., Zhu, Z.N.: Solving the \((3+1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Manafian, J., Mohammadi-Ivatloo, B., Abapour, M.: Lump-type solutions and interaction phenomenon to the \((2+1)\)-dimensional Breaking Soliton equation. Appl. Math. Comput. 356, 13–41 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Kumar, V., Gupta, R.K., Jiwari, R.: Lie group analysis, numerical and non-traveling wave solutions for the \((2+1)\)-dimensional diffusion-advection equation with variable coefficients. Chin. Phys. B 23(3), 030201 (2014)

    Article  Google Scholar 

  26. Gupta, R.K., Kumar, V., Jiwari, R.: Exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. Nonlinear Dyn. 79, 455–464 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kumar, S., Kumar, A.: Lie symmetry reductions and group invariant solutions of \((2+1)\)-dimensional modified Veronese web equation. Nonlinear Dyn. 98(3), 1891–1903 (2019)

    Article  MATH  Google Scholar 

  28. Kumar, S., Kumar, A., Wazwaz, A.M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135(11), 870 (2020)

    Article  Google Scholar 

  29. Kumar, S., Kumar, A., Kharbanda, H.: Lie symmetry analysis and generalized invariant solutions of \((2+1)\)-dimensional dispersive long wave (DLW) equations. Phys. Scr. 95(6), 065207 (2020)

    Article  Google Scholar 

  30. Jiwari, R., Kumar, V., Singh, S.: Lie group analysis, exact solutions and conservation laws to compressible isentropic Navier–Stokes equation. Eng. Comput. (2020) (in press). https://doi.org/10.1007/s00366-020-01175-9

  31. Osman, M.S., Baleanu, D., Adem, A.R., Hosseini, K., Mirzazadeh, M., Eslami, M.: Double-wave solutions and Lie symmetry analysis to the \((2+1)\)-dimensional coupled Burgers equations. Chin. J. Phys. 63, 122–129 (2020)

    Article  MathSciNet  Google Scholar 

  32. Kumar, S., Rani, S.: Lie symmetry reductions and dynamics of soliton solutions of \((2+1)\)-dimensional Pavlov equation. Pramana J. Phys. 94, 116 (2020)

    Article  Google Scholar 

  33. Zhao, D., Zhaqilao: Three-wave interactions in a more general \((2+1)\)-dimensional Boussinesq equation. Eur. Phys. J. Plus 135, 617 (2020)

    Article  Google Scholar 

  34. Han, P.F., Taogetusang: Higher-order mixed localized wave solutions and bilinear auto-Bäcklund transformations for the \((3+1)\)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Eur. Phys. J. Plus 137, 216 (2022)

    Article  Google Scholar 

  35. Nisar, K.S., Ilhan, O.A., Abdulazeez, S.T., Manafian, J., Mohammed, S.A., Osman, M.S.: Novel multiple soliton solutions for some nonlinear PDEs via multiple exp-function method. Results Phys. 21, 103769 (2021)

    Article  Google Scholar 

  36. Osman, M.S.: Nonlinear interaction of solitary waves described by multi-rational wave solutions of the \((2+1)\)-dimensional Kadomtsev–Petviashvili equation with variable coefficients. Nonlinear Dyn. 87, 1209–1216 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen, S.S., Tian, B., Qu, Q.X., Li, H., Sun, Y., Du, X.X.: Alfvén solitons and generalized Darboux transformation for a variable-coefficient derivative nonlinear Schrödinger equation in an inhomogeneous plasma. Chaos Solitons Fract. 148, 111029 (2021)

    Article  Google Scholar 

  38. Wang, M., Tian, B.: In an inhomogeneous multicomponent optical fiber: Lax pair, generalized Darboux transformation and vector breathers for a three-coupled variable-coefficient nonlinear Schrödinger system. Eur. Phys. J. Plus 136, 1002 (2021)

    Article  Google Scholar 

  39. Shen, Y., Tian, B.: Bilinear auto-Bäcklund transformations and soliton solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves. Appl. Math. Lett. 122, 107301 (2021)

    Article  MATH  Google Scholar 

  40. Gao, X.Y., Guo, Y.J., Shan, W.R.: Optical waves/modes in a multicomponent inhomogeneous optical fiber via a three-coupled variable-coefficient nonlinear Schrödinger system. Appl. Math. Lett. 120, 107161 (2021)

    Article  MATH  Google Scholar 

  41. Zhang, R.F., Li, M.C., Gan, J.Y., Li, Q., Lan, Z.Z.: Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos Solitons Fract. 154, 111692 (2022)

    Article  MathSciNet  Google Scholar 

  42. Zhang, R.F., Li, M.C.: Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn (2022) In press. https://doi.org/10.1007/s11071-022-07207-x

  43. Manafian, J., Mohammadi Ivatloo, B., Abapour, M.: Breather wave, periodic, and cross-kink solutions to the generalized Bogoyavlensky-Konopelchenko equation. Math. Meth. Appl. Sci. 43(4), 1753–1774 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)

    Article  MathSciNet  Google Scholar 

  45. Liu, J.G., Wazwaz, A.M.: Breather wave and lump-type solutions of new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid. Math. Meth. Appl. Sci. 44, 2200–2208 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  46. Chen, S.J., Lü, X., Tang, X.F.: Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients. Commun Nonlinear Sci Numer Simulat. 95, 105628 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gao, X.Y., Guo, Y.J., Shan, W.R.: Long waves in oceanic shallow water: Symbolic computation on the bilinear forms and Bäcklund transformations for the Whitham-Broer-Kaup system. Eur. Phys. J. Plus 135, 689 (2020)

    Article  Google Scholar 

  48. Gao, X.Y., Guo, Y.J., Shan, W.R.: Looking at an open sea via a generalized (2+1)-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons. Eur. Phys. J. Plus 136, 893 (2021)

    Article  Google Scholar 

  49. Gao, X.Y., Guo, Y.J., Shan, W.R.: Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq-Burgers system for the shallow water waves in a lake or near an ocean beach. Commun. Theor. Phys. 72, 095002 (2020)

    Article  MATH  Google Scholar 

  50. Gao, X.Y., Guo, Y.J., Shan, W.R.: Beholding the shallow water waves near an ocean beach or in a lake via a Boussinesq-Burgers system. Chaos Solitons Fract. 147, 110875 (2021)

    Article  MathSciNet  Google Scholar 

  51. Gao, X.Y., Guo, Y.J., Shan, W.R.: Symbolic computation on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system for the water waves. Chaos Solitons Fract. 150, 111066 (2021)

    Article  MathSciNet  Google Scholar 

  52. Liu, S.H., Tian, B., Wang, M.: Painlevé analysis, bilinear form, Bäcklund transformation, solitons, periodic waves and asymptotic properties for a generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma. Eur. Phys. J. Plus. 136, 917 (2021)

    Article  Google Scholar 

  53. Azmol, H.M., Ali, A.M., Shamim, S.S.: Abundant general solitary wave solutions to the family of KdV type equations. J. Ocean. Eng. Sci. 2, 47–54 (2017)

    Article  Google Scholar 

  54. Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: Quasi-periodic Waves and Solitary Waves to a Generalized KdV-Caudrey-Dodd-Gibbon Equation from Fluid Dynamics. Taiwan. J. Math. 20(4), 823–848 (2016)

    MathSciNet  MATH  Google Scholar 

  55. Kumar, M., Tiwari, A.K., Kumar, R.: Some more solutions of Kadomtsev-Petviashvili equation. Comput. Math. Appl. 74, 2599–2607 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  56. Chen, S.T., Ma, W.X.: Lump solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation. Comput. Math. Appl. 76, 1680–1685 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180–2186 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Han, P.F., Bao, T.: Integrability aspects and some abundant solutions for a new (4+1)-dimensional KdV-like equation. Int. J. Mod. Phys. B 35(6), 2150079 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhao, D., P. Zhaqilao.: The abundant mixed solutions of (2+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation. Nonlinear Dyn. 103, 1055-1070 (2021)

    Article  Google Scholar 

  62. Manafian, J., Lakestani, M.: N-lump and interaction solutions of localized waves to the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation. J. Geom. Phys. 150, 103598 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhao, Z.L., He, L.C.: M-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation. Appl. Math. Lett. 111, 106612 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  64. Tang, Y.N., Liang, Z.J., Ma, J.L.: Exact solutions of the (3+1)-dimensional Jimbo-Miwa equation via Wronskian solutions: Soliton, breather, and multiple lump solutions. Phys. Scr. 96, 095210 (2021)

    Article  Google Scholar 

  65. Feng, Y.Y., Wang, X.M., Bilige, S.D.: Evolutionary behavior and novel collision of various wave solutions to (3+1)-dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation. Nonlinear Dyn. 104, 4265–4275 (2021)

    Article  Google Scholar 

  66. Manafian, J., Lakestani, M.: Interaction among a lump, periodic waves, and kink solutions to the fractional generalized CBS-BK equation. Math. Meth. Appl. Sci. 44(1), 1052–1070 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  67. Wang, X., Wei, J., Geng, X.G.: Rational solutions for a (3+1)-dimensional nonlinear evolution equation. Commun Nonlinear Sci Numer Simul. 83, 105116 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  68. Han, P.F., Bao, T.: Bäcklund transformation and some different types of N-soliton solutions to the (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. Math. Meth. Appl. Sci. 44, 11307–11323 (2021)

    Article  MATH  Google Scholar 

  69. Manafian, J., Lakestani, M.: Lump-type solutions and interaction phenomenon to the bidirectional Sawada-Kotera equation. Pramana-J Phys. 92, 41 (2019)

    Article  Google Scholar 

  70. Liu, J.G., Osman, M.S., Zhu, W.H., Zhou, L., Baleanu, D.: The general bilinear techniques for studying the propagation of mixed-type periodic and lump-type solutions in a homogenous-dispersive medium. AIP Adv. 10, 105325 (2020)

    Article  Google Scholar 

  71. Osman, M.S., Wazwaz, A.M.: A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Math. Meth. Appl. Sci. 42, 6277–6283 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  72. Han, P.F., Bao, T.: Interaction of multiple superposition solutions for the (4+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Nonlinear Dyn. 105, 717–734 (2021)

    Article  Google Scholar 

  73. Miao, Z.W., Hu, X.R., Chen, Y.: Interaction phenomenon to (1+1)-dimensional Sharma-Tasso-Olver-Burgers equation. Appl. Math. Lett. 112, 106722 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  74. Hirota R. The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  75. Manafian, J., Ilhan, O.A., Avazpour, L., Alizadeh, A.: N-lump and interaction solutions of localized waves to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation arise from a model for an incompressible fluid. Math. Meth. Appl. Sci. 43, 9904–9927 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors deeply appreciate the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further. This work has been supported by the National Natural Science Foundation of China (Grant No. 11361040), the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2020LH01008), the Graduate Students’ Scientific Research Innovation Fund Program of Inner Mongolia Normal University, China (Grant Nos. CXJJS19096, CXJJS20089) and the Graduate Research Innovation Project of Inner Mongolia Autonomous Region, China (Grant No. S20191235Z).

Author information

Authors and Affiliations

  1. College of Mathematics Science, Inner Mongolia Normal University, Huhhot, 010022, People’s Republic of China

    Peng-Fei Han & Taogetusang Bao

  2. Center for Applied Mathematics Inner Mongolia, Huhhot, 010022, People’s Republic of China

    Peng-Fei Han & Taogetusang Bao

Authors
  1. Peng-Fei Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taogetusang Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taogetusang Bao.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the research effort and the publication of this paper.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, PF., Bao, T. Hybrid localized wave solutions for a generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system in a fluid or plasma. Nonlinear Dyn 108, 2513–2530 (2022). https://doi.org/10.1007/s11071-022-07327-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07327-4

Keywords

Search

Navigation