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Bell polynomials and lump-type solutions to the Hirota–Satsuma–Ito equation under general and positive quadratic polynomial functions

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Abstract

In this article, we explore the Hirota–Satsuma–Ito (HSI) equation in (2+1)-dimensions which possess lump solutions. We first used the concept of Bell’s polynomials to derive the bilinear form of the equation. Then, we proceed to derive a quadratic function solution of the bilinear form and then expand it as the sums of squares of linear functions satisfying some conditions. Most importantly, we acquire a lump-type solution containing 11 parameters along with some non-zero conditions necessary for the existence of the solutions. Then, lump solutions are derived from the from the lump-type solutions by choosing a set of the constant. The solutions obtained in this paper further enrich the literature the ones reported in previous time using different Hirota bilinear approaches and the category of nonlinear partial differential equations (NPDEs) which possess lump solutions, particularly the HSI equation. The physical interpretation of the results is discussed and represented graphically.

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References

  1. 1.

    G.B. Whitham, Linear and Nonlinear Waves (John Whiley, New york, 1974)

  2. 2.

    A. Hesegawa, Y. Kodama, Solitons in Optical Communication (Oxford University Press, Oxford, 1995)

  3. 3.

    Q. Li, T. Chaolu, Y.H. Wanga, Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation. Computers and Mathematics with Applications 77, 2077–2085 (2019)

  4. 4.

    X. Yong, X. Li, Y. Huang, General lump-type solutions of the (3+1)-dimensional Jimbo-Miwa equation. Applied Mathematics Letters 86, 222–228 (2018)

  5. 5.

    W.X. Ma, Z.Y. Qin, X. Lu, Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dynam 84, 923–931 (2016)

  6. 6.

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

  7. 7.

    H. Wang, Y.H. Wang, W.X. Ma, C. Temuer, Lump solutions of a new extended (2 + 1)-dimensionalBoussinesq equation. Mod Phys Lett B 32, 1850376 (2018)

  8. 8.

    H. Wang, Lump and interaction solutions to the (2 + 1)-dimensional Burgers equation. Appl Math Lett 85, 27–34 (2018)

  9. 9.

    W.X. Ma, Y. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans. Amer. Math. Soc. 357, 1753–1778 (2005)

  10. 10.

    W.X. Ma, A search for lump solutions to a combined fourth-order nonlinear PDE in (2+1)-dimension, Journal of Applied Analysis and Computation. Article in press. 2019. DOI:https://doi.org/10.11948/*.1

  11. 11.

    J. Hietarinta, Introduction to the Hirota bilinear method, in Integrability of Nonlinear Systems, ed. by Y. Kosmann-Schwarzbach, B. Grammaticos, K.M. Tamizhmani (Springer, Berlin, Heidelberg, 1997), pp. 95–103

  12. 12.

    Y. Zhou, S. Manukure, W.X. Ma, Lump and lump-soliton solutions to the Hirota-Satsuma-Ito equation. Commun Nonlinear Sci Numer Simulat 68, 56–62 (2019)

  13. 13.

    W.X. Ma, J. Li, C.M. Khaliq, A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions. Complexity 2018, 9059858 (2018)

  14. 14.

    E.T. Bell, Exponential polynomials. Ann. Math 35, 258–277 (1934)

  15. 15.

    T.T. Zhang, P.L. Ma, M.J. Xu, X.Y. Zhang, S.F. Tian, On Bell polynomials approach to the integrability of a (3 + 1)-dimensional generalized Kadomtsev Petviashvili equation. Modern Physics Letters B 29, 1550051 (2015)

  16. 16.

    G. Gilson, F. Lambert, J.J.C. Nimmo, R. Willox, On the combinatorics of the Hirota D-operators. Proc. R. Soc. Lond. A 452, 223–234 (1996)

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11571378).

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Correspondence to Yongjin Li.

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The authors declare that they have no conflict of interest. All the authors have read and approved the final draft of the paper

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Aliyu, A.I., Li, Y. Bell polynomials and lump-type solutions to the Hirota–Satsuma–Ito equation under general and positive quadratic polynomial functions. Eur. Phys. J. Plus 135, 119 (2020). https://doi.org/10.1140/epjp/s13360-019-00054-7

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