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Nonlinear superposition of the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

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Abstract

In this work, the (2+1)-dimensional gKDKK equation is the research object. Different from previous studies, this study aims to obtain the nonlinear superposition of the (2+1)-dimensional gKDKK equation by adding new constraints, which leads to new results of its solution states. The non-collision of lump wave with line wave and breather wave is studied. The results show that the two conditions satisfy the requirement that lump wave does not collide with line wave and lump wave does not collide with breather wave. At the same time, the mixed solutions of lump wave, line wave and breather wave are also studied and the same conclusion is obtained. The results of this nonlinear superposition will break the traditional research and further enrich the dynamic behaviors of nonlinear evolution equations.

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References

  1. Jia, M., Lin, J., Lou, S.: Soliton and breather molecules in few-cycle-pulse optical model. Nonlinear Dyn. 100(4), 3745–3757 (2020)

    Article  Google Scholar 

  2. Yan, Z., Lou, S.: Special types of solitons and breather molecules for a (2+ 1)-dimensional fifth-order KdV equation. Commun. Nonlinear Sci. Numer. Simul. 91, 105425 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ma, W.: N-soliton solution and the hirota condition of a (2+1)-dimensional combined equation. Math. Comput. Simul. 190, 270–279 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wazwaz, A.-M.: Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations. Appl. Math. Lett. 64, 21–26 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dong, J., Li, B., Yuen, M.: Soliton molecules and mixed solutions of the (2+1)-dimensional bidirectional Sawada–Kotera equation. Commun. Theor. Phys. 72(2), 025002 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  7. Yan, Z., Lou, S.: Soliton molecules in Sharma-Tasso-Olver-Burgers equation. Appl. Math. Lett. 104, 106271 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang, X., Fan, R., Li, B.: Soliton molecules and some novel interaction solutions to the (2+1)-dimensional b-type Kadomtsev-Petviashvili equation. Phys. Scr. 95(4), 045213 (2020)

    Article  Google Scholar 

  9. Xu, D., Lou, S.: Dark soliton molecules in nonlinear optics. Acta Physica Sinica 69(1), 20191347 (2020)

  10. Zhang, Z., Yang, S., Li, B.: Soliton molecules, asymmetric solitons and hybrid solutions for (2+1)-dimensional fifth-order KdV equation. Chin. Phys. Lett. 36(12), 120501 (2019)

    Article  Google Scholar 

  11. Chow, K.W., Grimshaw, R.H.J., Ding, E.: Interactions of breathers and solitons in the extended Korteweg-de Vries equation. Wave Motion 43(2), 158–166 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhu, J., Wang, B., Ma, Z., Fei, J.: Soliton molecules and some related interaction solutions of the (2+1)-dimensional Kadomtsev-Petviashvili hierarchy. Mod. Phys. Lett. B 35(06), 2150106 (2021)

    Article  MathSciNet  Google Scholar 

  13. Li, J., Chen, Q., Li, B.: Resonance y-type soliton solutions and some new types of hybrid solutions in the (2+1)-dimensional Sawada-Kotera equation. Commun. Theor. Phys. 73(4), 045006 (2021)

    Article  MathSciNet  Google Scholar 

  14. Zhao, Z., He, L.: Resonance y-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. Appl. Math. Lett. 122, 107497 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, M., Qi, Z., Chen, J., Li, B.: Resonance y-shaped soliton and interaction solutions in the (2+1)-dimensional b-type Kadomtsev-Petviashvili equation. Int. J. Mod. Phys. B 35(21), 2150222 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ma, H., Huang, H., Deng, A.: Solitons and soliton molecules in two nonlocal Alice-Bob fifth-order KdV systems. Int. J. Theor. Phys. 60(8), 3051–3062 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, Z., Guo, Q., Li, B., Chen, J.: A new class of nonlinear superposition between lump waves and other waves for Kadomtsev-Petviashvili I equation. Commun. Nonlinear Sci. Numer. Simul. 101, 105866 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhao, Z., He, L.: Nonlinear superposition between lump waves and other waves of the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. Nonlinear Dyn. 108(1), 555–568 (2022)

    Article  Google Scholar 

  19. 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(2), 1617–1626 (2022)

    Article  Google Scholar 

  20. Li, L., Gao, Y., Lei, H., Jia, T., Ding, C., Feng, Y.: Bilinear form, soliton, breather, lump and hybrid solutions for a (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 100(3), 2729–2738 (2020)

    Article  Google Scholar 

  21. Zhang, H., Ma, W.: Lump solutions to the (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 87(4), 2305–2310 (2017)

    Article  MathSciNet  Google Scholar 

  22. Xin, X., Liu, X., Zhang, L.: Explicit solutions of the Bogoyavlensky-Konoplechenko equation. Appl. Math. Comput. 215(10), 3669–3673 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Lü, X., Li, J.: Integrability with symbolic computation on the Bogoyavlensky-Konoplechenko model: Bell-polynomial manipulation, bilinear representation, and wronskian solution. Nonlinear Dyn. 77(1), 135–143 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gu, Y.: Analytical solutions to the Caudrey–Dodd–Gibbon–Sawada–Kotera equation via symbol calculation approach. J. Funct. Spaces 2020, 1 (2020)

  25. Deng, Z., Chang, X., Tan, J., Tang, B., Deng, K.: Characteristics of the lumps and stripe solitons with interaction phenomena in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation. Int. J. Theor. Phys. 58(1), 92–102 (2019)

  26. Feng, L., Tian, S., Yan, H., Wang, L., Zhang, T.: On periodic wave solutions and asymptotic behaviors to a generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Eur. Phys. J. Plus 131(7), 1–18 (2016)

  27. Zhou, X., Ilhan, O.A., Manafian, J., Singh, G., Tuguz, N.S.: N-lump and interaction solutions of localized waves to the (2+1)-dimensional generalized KDKK equation. J. Geom. Phys. 168, 104312 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, W., Zhang, Y., Shi, D.: Analysis on lump, lumpoff and rogue waves with predictability to a generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Commun. Theor. Phys. 71(6), 670 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ma, H., Cheng, Q., Deng, A.: Soliton molecules and some novel hybrid solutions for the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Commun. Theor. Phys. 72(9), 095001 (2020)

    Article  MathSciNet  Google Scholar 

  30. Fan, S., Wu, H., Fei, J., Cao, W., Ma, Z.: Soliton molecule and their interaction solutions for the (2+1)-dimensional gKDKK equation. Int. J. Mod. Phys. B 2250048 (2022)

  31. Ma, H., Gao, Y., Deng, A.: Fission and fusion solutions of the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation: case of fluid mechanics and plasma physics. Nonlinear Dyn. 108(4), 4123–4137 (2022)

    Article  Google Scholar 

  32. Zhang, Z., Yang, X., Li, W., Li, B.: Trajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev-Petviashvili equation. Chin. Phys. B 28(11), 110201 (2019)

    Article  Google Scholar 

  33. Ren, B.: Dynamics of a D’alembert wave and a soliton molecule for an extended BLMP equation. Commun. Theor. Phys. 73(3), 035003 (2021)

    Article  MathSciNet  Google Scholar 

  34. Man, J., Lou, S.: Searching for missing d’alembert waves in nonlinear system: Nizhnik–Novikov–Veselov equation. Chaos Solitons Fract. 140, 110135 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ma, H., Yue, S., Deng, A.: D’alembert wave, the hirota conditions and soliton molecule of a new generalized KdV equation. J. Geom. Phys. 172, 104413 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ma, H., Gao, Y., Deng, A.: D’alembert wave and soliton molecule of the generalized Nizhnik–Novikov–Veselov equation. Mod. Phys. Lett. B 35(31), 2150482 (2021)

    Article  MathSciNet  Google Scholar 

  37. Zhang, X., Chen, Y., Tang, X.: Rogue wave and a pair of resonance stripe solitons to KP equation. Comput. Math. Appl. 76(8), 1938–1949 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhao, Z., He, L., Gao, Y.: Rogue wave and multiple lump solutions of the (2+1)-dimensional Benjamin-Ono equation in fluid mechanics. Complexity 2019, 1 (2019)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  40. He, L., Zhang, J., Zhao, Z.: Resonance y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2 +1)-dimensional nonlinear wave equation. Nonlinear Dyn. 106(3), 2515–2535 (2021)

  41. Qi, Z., Zhang, Z., Li, B.: Space-curved resonant line solitons in a generalized (2+1)-dimensional fifth-order KdV system. Chin. Phys. Lett. 38(6), 060501 (2021)

  42. Ma, H., Yue, S., Deng, A.: Resonance y-shape solitons and mixed solutions for a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Sawada-Kotera equation in fluid mechanics. Nonlinear Dyn. 108(1), 505–519 (2022)

  43. Tang, X., Cui, C., Liang, Z., Ding, W.: Novel soliton molecules and wave interactions for a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 105(3), 2549–2557 (2021)

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Authors and Affiliations

  1. Department of Applied Mathematics, Donghua University, Shanghai, 201620, China

    Hongcai Ma, Yidan Gao & Aiping Deng

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  1. Hongcai Ma
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  2. Yidan Gao
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Correspondence to Hongcai Ma.

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Ma, H., Gao, Y. & Deng, A. Nonlinear superposition of the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Nonlinear Dyn 111, 619–632 (2023). https://doi.org/10.1007/s11071-022-07827-3

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