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Some novel fusion and fission wave solutions in the \((2\!+\!1)\)-dimensional generalized Bogoyavlensky–Konopelchenko equation

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Abstract

The phenomena of fusion and fission are important, which have been experimentally observed in many real physical models. In this paper, by introducing a new constraint to the N-solitons, we obtain some novel fusion and fission wave solutions for the \((2+1)\)-dimensional generalized BK equation. Numerical simulations reveal that the fusion and fission wave solutions take the shape of Y in the spatial structure. Moreover, we construct two different kinds of interaction solutions: One is the hybrid solution mixed by the fusion (fission) solitons and T-order breathers by using the constraint and complexification method. The other one is the hybrid solution consisting of the fusion (fission) solitons and L-order lumps by utilizing the constraint and long wave limit approach. Dynamical behaviors and mathematical features of these hybrid solutions are analyzed theoretically and graphically. What needs to be pointed out is that our method constitutes a generalization of what was given by Chen and Wang et al., which can be effectively used to construct the fusion and fission wave solutions of other nonlinear PDEs. Based on the wide applications of fusion and fission phenomena in nonlinear optics, fluid, biology, oceanic and atmospheric dynamics, the solutions obtained here may be helpful for experts to explain or predict some related physical phenomena.

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We declare that the data supporting the findings of this study are available within the article.

References

  1. M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting (Cambridge University Press, NewYork, 1991)

    Book  Google Scholar 

  2. C. Rogers, W. Schief, Bäcklund and Darboux Transformations Geometry and Modern Applications in Soliton Theory (Cambridge University Press, NewYork, 2002)

    Book  MATH  Google Scholar 

  3. V.A. Matveev, M.A. Salle, Darboux Transformations and Solitons, Berlin (Springer-Verlag, Heidelberg, 1991)

    Book  MATH  Google Scholar 

  4. Q.P. Liu, M. Manas, Darboux transformations for super-symmetric KP hierarchies. Phys. Lett. B. 485, 293–300 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer-Verlag, New York, 1989)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  7. X.B. Hu, Generalized Hirota’s bilinear equations and their soliton solutions. J. Phys. A Math. Gen. 26, L465–L471 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. X.D. Shen, J. Manafian, M. Jiang, O.A. Ilhan, S.S. Shafik, M. Zaidi, Abundant wave solutions for generalized Hietarinta equation with Hirota’s bilinear operator. Mod. Phys. Lett. B 36, 2250032 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  9. J. Manafian, O.A. Ilhan, A. Alizadeh, S.A. Mohammed, Multiple rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics. Commun. Theor. Phys. 72, 075002 (13pp) (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. X. Hong, A. Alkireet, O.A. Ilhan, J. Manafian, M. Nasution, Multiple soliton solutions of the generalized Hirota-Satsuma-Ito equation arising in shallow water wave. J. Geom. Phys. 170, 104338–104356 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. W.X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82(6), 065003, 8pp (2010)

    Article  ADS  MATH  Google Scholar 

  12. Y.L. Ma, A.M. Wazwaz, B.Q. Li, A new (3+1)-dimensional Kadomtsev-Petviashvili equation and its integrability, multiple-solitons, breathers and lump waves. Math. Comput. Simul. 187, 505–519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. G.P. Shen, J. Manafian, D.T.N. Huy, K.S. Nisar, M. Abotaleb, N.D. Trung, Abundant soliton wave solutions and the linear superposition principle for generalized (3+1)-D nonlinear wave equation in liquid with gas bubbles by bilinear analysis. Results Phys. 32, 105066 (2022)

    Article  Google Scholar 

  14. P. Kumari, R.K. Gupta, S. Kumar, K.S. Nisar, Doubly periodic wave structure of the modified Schrödinger equation with fractional temporal evolution. Results Phys. 33, 105128 (2022)

    Article  Google Scholar 

  15. A. Zafar, M. Raheel, M.Q. Zafar, K.S. Nisar, M.S. Osman, R.N. Mohamed, A. Elfasakhany, Dynamics of different nonlinearities to the perturbed nonlinear Schrödinger equation via solitary wave solutions with numerical simulation. Fractal Fract. 5, 213 (2021)

    Article  Google Scholar 

  16. A. Hussain, A. Jhangeer, N. Abbas, I. Khan, K.S. Nisar, Solitary wave patterns and conservation laws of fourth-order nonlinear symmetric regularized long-wave equation arising in plasma. Ain Shams Eng. J. 12, 3919–3930 (2021)

    Article  Google Scholar 

  17. M. Rani, N. Ahmed, S.S. Dragomir, S.T. Mohyud-Din, I. Khan, K.S. Nisar, Some newly explored exact solitary wave solutions to nonlinear inhomogeneous Murnaghan’s rod equation of fractional order. J. Taibah Univ. Sci. 15, 97–110 (2021)

    Article  Google Scholar 

  18. A. Jhangeer, A. Hussain, M. Junaid-U-Rehman, I. Khan, D. Baleanu, K.S. Nisar, Lie analysis, conservation laws and travelling wave structures of nonlinear Bogoyavlenskii–Kadomtsev–Petviashvili equation. Results Phys. 19, 103492 (2020)

    Article  Google Scholar 

  19. M. Hisakado, Breather trapping mechanism in piecewise homogeneous DNA. Phys. Lett. A 227, 87–93 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. D. Kip, M. Wesner, C. Herden, V. Shandarov, Interaction of spatial photorefractive solitons in a planar waveguide. Appl. Phys. B. 68, 971–974 (1999)

    Article  ADS  Google Scholar 

  21. V.N. Serkin, V.M. Chapela, J. Percino, T.L. Belyaeva, Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides. Opt. Commun. 192, 237–244 (2001)

    Article  ADS  Google Scholar 

  22. G. Stoitcheva, L. Ludu, J. Draayer, Antisoliton model for fission modes. Math. Comput. Simul. 55, 621–625 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. A.H. Chen, Multi-kink solutions and soliton fission and fusion of Sharma–Tasso–Olver equation. Phys. Lett. A 374, 2340–2345 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. C.K. Kuo, W.X. Ma, A study on resonant multi-soliton solutions to the (2+1)-dimensional Hirota-Satsuma-Ito equations via the linear superposition principle. Nonlinear Anal. 190, 111592 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Y.F. Wang, B. Tian, Y. Jiang, Soliton fusion and fission in a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids. Appl. Math. Comput. 292, 448–456 (2017)

    MathSciNet  MATH  Google Scholar 

  27. W. Liu, X.X. Zheng, C. Wang, S.Q. Li, Fission and fusion collision of high-order lumps and solitons in a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 96, 2463–2473 (2019)

    Article  MATH  Google Scholar 

  28. A.H. Chen, F.F. Wang, Fissionable wave solutions, lump solutions and interactional solutions for the (2+1)-dimensional Sawada–Kotera equation. Phys. Scr. 94, 055206, 8pp (2019)

    Article  Google Scholar 

  29. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectrum Transform Method (World Scientific, Singapore, 1993)

    Book  MATH  Google Scholar 

  30. M.V. Prabhakar, H. Bhate, Exact solutions of the Bogoyavlensky–Konoplechenko equation. Lett. Math. Phys. 64, 1–6 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. S. Saha Ray, On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation. Comput. Math. Appl 74, 1158–1165 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Triki, Z. Jovanoski, A. Biswas, Shock wave solutions to the Bogoyavlensky–Konopelchenko equation. Indian J. Phys 88, 71–74 (2014)

    Article  ADS  Google Scholar 

  33. K. Toda, S.J. Yu, A study of the construction of equations in (2+1)-dimensions. Inverse Probl. 17, 1053–1060 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. A..M. Wazwaz, The (2+1) and (3+1)-dimensional CBS equations: multiple soliton solutions and multiple singular soliton solutions. Z. Naturforsch 65a, 173–181 (2010)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. X. Lü, J. Li, Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation, bilinear representation, and Wronskian solution. Nonlinear Dyn. 77, 135–143 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. H. Yan, S.F. Tian, L.L. Feng, T.T. Zhang, Quasi-periodic wave solutions, soliton solutions, and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation. Waves Random Complex Media 26, 444–457 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Q. Li, T. Chaolu, Y.H. Wang, Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. Comput. Math. Appl. 77, 2077–2085 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. H.C. Ma, Y.X. Bai, A.P. Deng, General M-lump, high-order breather, and localized interaction solutions to (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation. Front. Math. China (2021). https://doi.org/10.1007/s11464-021-0918-5

    Article  Google Scholar 

  41. X.M. Zhou, S.F. Tian, L.D. Zhang, T.T. Zhang, General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. Mod. Phys. Lett. B 35, 2150057 (2020)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

We would like to express our sincere thanks to the reviewers for their helpful comments and valuable suggestions. This work is supported by Jiangsu Provincial Natural Science Foundation under Grant No. BK20221508, the National Natural Science Foundation of China under Grant No. 11775116, No. BK20210380, JSSCBS20210277 and Jiangsu Qinglan high-level talent Project.

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Authors and Affiliations

  1. College of Sciences, Nanjing Agricultural University, Nanjing, People’s Republic of China

    Yuhan Li & Hongli An

  2. College of Economics and Management, Nanjing Forestry University, Nanjing, People’s Republic of China

    Haixing Zhu

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  1. Yuhan Li
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Correspondence to Hongli An.

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Li, Y., An, H. & Zhu, H. Some novel fusion and fission wave solutions in the \((2\!+\!1)\)-dimensional generalized Bogoyavlensky–Konopelchenko equation. Eur. Phys. J. Plus 137, 1384 (2022). https://doi.org/10.1140/epjp/s13360-022-03605-7

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