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A new type of multiple-lump and interaction solution of the Kadomtsev–Petviashvili I equation

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Abstract

In this paper, the solution in the form of Grammian of the Kadomtsev–Petviashvili I equation is employed to investigate a new type of multiple-lump solution. The bound state called lump molecules appears in a period of time. A generalized form of the N-lump solutions in N-order determinant possessing the structure of lump molecules is explicitly given. Utilizing the nonzero constant matrices and a non-homogeneous polynomial, the interaction solutions between the multiple-lump waves and the line solitons are constructed. The interactions among the N-lumps and between lumps and solitons are investigated with the aid of numerical simulation. The results extend the understanding of the multiple lumps and interaction dynamical behaviors of the Kadomtsev–Petviashvili I equation.

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References

  1. He, J.S., Xu, S.W., Porsezian, K., Cheng, Y., Dinda, P.T.: Rogue wave triggered at a critical frequency of a nonlinear resonant medium. Phys. Rev. E 93, 062201 (2016)

    Article  MathSciNet  Google Scholar 

  2. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  3. Chen, J.B., Pelinovsky, D.E., White, R.E.: Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation. Phys. Rev. E 100, 052219 (2019)

    Article  MathSciNet  Google Scholar 

  4. Zhang, X., Wang, L., Liu, C., Li, M., Zhao, Y.C.: High-dimensional nonlinear wave transitions and their mechanisms. Chaos 30, 113107 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yuan, F., Cheng, Y., He, J.S.: Degeneration of breathers in the Kadomttsev–Petviashvili I equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105027 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wazwaz, A.M.: Solitary waves theory. In: Luo, A.C.J., Ibragimov, N.H. (eds.) Partial Differential Equations and Solitary Waves Theory. Nonlinear Physical Science. Springer, Berlin (2009)

  7. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fokas, A.S., Pelinovsky, D.E., Sulem, C.: Interaction of lumps with a line soliton for the DSII equation. Phys. D 152–153, 189–198 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264, 2633–2659 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Singh, N., Stepanyants, Y.: Obliquely propagating skew KP lumps. Wave Motion 64, 92–102 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Zhao, Z.L., He, L.C.: M-lump, high-order breather solutions and interaction dynamics of a generalized (2+1)-dimensional nonlinear wave equation. Nonlinear Dyn. 100, 2753–2765 (2020)

    Article  Google Scholar 

  13. He, L.C., Zhang, J.W., Zhao, Z.L.: M-lump and interaction solutions of a (2+1)-dimensional extended shallow water wave equation. Eur. Phys. J. Plus 136, 192 (2021)

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Sun, Y.L., Chen, J., Ma, W.X., Yu, J.P., Khalique, C.M.: Further study of the localized solutions of the (2+1)-dimensional B-Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 107, 106131 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao, Z.L., He, L.C.: 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 

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

  18. Matveev, B.B.: Darboux transformation and explicit solutions of the Kadomtcev–Petviaschvily equation, depending on functional parameters. Lett. Math. Phys. 3, 213–216 (1979)

    Article  MathSciNet  Google Scholar 

  19. He, J.S., Cheng, Y., Li, Y.S.: The Darboux transformation for NLS-MB equations. Commun. Theor. Phys. 38, 493–496 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feng, B.F., Luo, X.D., Ablowitz, M.J., Musslimani, Z.H.: General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. Nonlinearity 31, 5385–5409 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rao, J.G., Chow, K.W., Mihalache, D., He, J.S.: Completely resonant collision of lumps and line solitons in the Kadomtsev–Petviashvili I equation. Stud. Appl. Math. 147, 1007–1035 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, B., Chen, J.C., Yang, J.K.: Rogue waves in the generalized derivative nonlinear Schödinger equations. J. Nonlinear Sci. 30, 3027–3056 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump and interactive solutions to a (3+1)-dimensional nonlinear system. Nonlinear Dyn. 93, 2533–2541 (2018)

    Article  MATH  Google Scholar 

  25. Ma, Y.L., Wazwaz, A.M., Li, B.Q.: New extended Kadomtsev–Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions. Nonlinear Dyn. 104, 1581–1594 (2021)

    Article  Google Scholar 

  26. Fokas, A.S., Ablowitz, M.J.: The inverse scattering transform for the Benjamin–Ono equation—a pivot to multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Olver, P.J.: Application of Lie Groups to Differential Equations. Springer (1993)

  28. Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer (2002)

  29. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer (2010)

  30. Zhao, Z.L., Han, B.: Lie symmetry analysis of the Heisenberg equation. Commun. Nonlinear Sci. Numer. Simul. 45, 220–234 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao, Z.L., He, L.C.: Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation. Theor. Math. Phys. 206, 142–162 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhao, Z.L.: Conservation laws and nonlocally related systems of the Hunter–Saxton equation for liquid crystal. Anal. Math. Phys. 9, 2311–2327 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhao, Z.L.: Symmetry-preserving difference models of some high-order nonlinear integrable equations. J. Nonlinear Math. Phys. 28, 452–465 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhao, Z.L., Han, B.: Lie symmetry analysis, Bäcklund transformations, and exact solutions of a (2+1)-dimensional Boiti–Leon–Pempinelli system. J. Math. Phys. 58, 101514 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, R.F., Li, M.C.: Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn. 108, 521–531 (2022)

    Article  Google Scholar 

  36. 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. 158, 111939 (2022)

    MathSciNet  Google Scholar 

  37. 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 

  38. Xu, J., Fan, E.G.: Long-time asymptotic behavior for the complex short pulse equation. J. Differ. Equ. 269, 10322–10349 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Guo, N., Xu, J., Wen, L.L., Fan, E.G.: Rogue wave and multi-pole solutions for the focusing Kundu–Eckhaus equation with nonzero background via Riemann–Hilbert problem method. Nonlinear Dyn. 103, 1851–1868 (2021)

    Article  Google Scholar 

  40. Chen, Y.Q., Tian, B., Qu, Q.X., Sun, Y., Chen, S.S., Hu, C.C.: Painlevé integrable condition, auto-Bäcklund transformations, Lax pair, breather, lump-periodic-wave and kink-wave solutions of a (3+1)-dimensional Hirota–Satsuma–Ito-like system for the shallow water waves. Nonlinear Dyn. 106, 765–773 (2021)

    Article  Google Scholar 

  41. Shen, Y., Tian, B., Liu, S.H.: Solitonic fusion and fission for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Phys. Lett. A 405, 127429 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Z., Yang, X.Y., Li, W.T., 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, 110201 (2019)

    Article  Google Scholar 

  43. Liu, Y.K., Li, B., An, H.L.: General high-order breathers, lumps in the (2+1)-dimensional Boussinesq equation. Nonlinear Dyn. 92, 2061–2076 (2018)

    Article  Google Scholar 

  44. Xu, G.Q., Wazwaz, A.M.: Integrability aspects and localized wave solutions for a new (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 98, 1379–1390 (2019)

    Article  Google Scholar 

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

    Article  Google Scholar 

  46. Li, Y., Yao, R.X., Xia, Y.R., Lou, S.Y.: Plenty of novel interaction structures of soliton molecules and asymmetric solitons to (2+1)-dimensional Sawada–Kotera equation. Commun. Nonlinear Sci. Numer. Simul. 100, 105843 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lv, N.N., Huang, L.: Breather-soliton molecules and breather-positons for the extended complex modified KdV equation. Commun. Nonlinear Sci. Numer. Simul. 107, 106148 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hao, X.Z.: Nonlocal symmetries and molecule structures of the KdV hierarchy. Nonlinear Dyn. 104, 4277–4291 (2021)

    Article  Google Scholar 

  49. Zhang, Z., Guo, Q., Li, B., Chen, J.C.: 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 

  50. Zhang, Z., Li, B., Wazwaz, A.M., Guo, Q.: Lump molecules in fluid systems: Kadomtsev–Petviashvili I case. Phys. Lett. A 424, 127848 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  51. Stepanyants, Y.A., Zakharov, D.V., Zakharov, V.E.: Lump interactions with plane solitons. arXiv:2108.06071, (2021)

  52. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970)

    MATH  Google Scholar 

  53. Zhou, X.: Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation. Commun. Math. Phys. 128, 551–564 (1990)

    Article  MATH  Google Scholar 

  54. Minzoni, A.A., Smyth, N.F.: Evolution of lump solutions for the KP equation. Wave Motion 24, 291–305 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  55. Lou, S.Y., Hu, X.B.: Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys. 38, 6401–6427 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  56. Deng, S.F., Chen, D.Y., Zhang, D.J.: The multisoliton solutions of the KP equation with self-consistent sources. J. Phys. Soc. Jpn. 72, 2184–2192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wazwaz, A.M.: Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Appl. Math. Comput. 190, 633–640 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Guo, L.J., Chabchoub, A., He, J.S.: Higher-order rogue wave solutions to the Kadomtsev–Petviashvili 1 equation. Phys. D 426, 132990 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  60. Liu, P., Cheng, J., Ren, B., Yang, J.R.: Bäcklund transformations, consistent Riccati expansion solvability, and soliton–cnoidal interaction wave solutions of Kadomtsev–Petviashvili equation. Chin. Phys. B 29, 020201 (2020)

    Article  Google Scholar 

  61. Lester, C., Gelash, A., Zakharov, D., Zakharov, V.: Lump chains in the KP-I equation. Stud. Appl. Math. 147, 1425–1442 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12101572), Shanxi Province Science Foundation for Youths (No. 201901D211274), Shanxi Province Science Foundation (20210302123019), Research Project Supported by Shanxi Scholarship Council of China (No. 2020-105) and the Fund for Shanxi “1331KIRT.”

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  1. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Zhonglong Zhao

  2. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, People’s Republic of China

    Lingchao He

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  1. Zhonglong Zhao
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Correspondence to Zhonglong Zhao.

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Zhao, Z., He, L. A new type of multiple-lump and interaction solution of the Kadomtsev–Petviashvili I equation. Nonlinear Dyn 109, 1033–1046 (2022). https://doi.org/10.1007/s11071-022-07484-6

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