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M-lump and interactive solutions to a (3 \({+}\) 1)-dimensional nonlinear system

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Abstract

This paper aims at computing M-lump solutions for the \((3+1)\)-dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state obtained by employing the “long wave” limit with respect to the N-soliton solutions which are got by using the direct methods. Subsequently, we discuss the dynamic properties of the M-lump solutions which describe the multiple collisions of lumps. Based on the obtained lump solutions, the lump–kink solutions are also obtained. In addition, the periodic interactive solutions are given.

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References

  1. Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Phys. Lett. A 63(3), 205–206 (1977)

    Article  Google Scholar 

  2. Krichever, I.M.: Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of n particles on a line. Funct. Anal. Appl. 12(1), 59–61 (1978)

    Article  MATH  Google Scholar 

  3. Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20(7), 1496–1503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Villarroel, J., Ablowitz, M.J.: On the discrete spectrum of the nonstationary schrödinger equation and multipole lumps of the Kadomtsev–Petviashvili i equation. Commun. Math. Phys. 207(1), 1–42 (1999)

    Article  MATH  Google Scholar 

  5. Imai, K.: Dromion and lump solutions of the Ishimori-i equation. Prog. Theor. Phys. 98(5), 1013–1023 (1997)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Lv, J.Q., Bilige, S.D.: Lump solutions of a (\(2+1\))-dimensional bsk equation. Nonlinear Dyn. 90(3), 2119–2124 (2017)

    Article  MathSciNet  Google Scholar 

  8. Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ma, W.X., Qin, Z.Y., Xing, L.: Lump solutions to dimensionally reduced p-gkp and p-gbkp equations. Nonlinear Dyn. 84(2), 923–931 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ma, W.X., Zhou, Y., Dougherty, R.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Mod. Phys. B 30(28n29), 1640018 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sun, H.Q., Chen, A.H.: Lump and lump-kink solutions of the (\(3+1\))-dimensional Jimbo–Miwa and two extended Jimbo–Miwa equations. Appl. Math. Lett. 68, 55–61 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, X.E., Chen, Y.: Deformation rogue wave to the (\(2+1\))-dimensional KdV equation. Nonlinear Dyn. 1, 755–763 (2017)

    MathSciNet  Google Scholar 

  14. Yan, Z.Y.: New families of nontravelling wave solutions to a new (\(3+1\))-dimensional potential-YTSF equation. Phys. Lett. A 318(12), 78–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wazwaz, A.M.: Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203(2), 592–597 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Zhang, T.X., Xuan, H.N., Zhang, D.F., Wang, C.J.: Non-travelling wave solutions to a (\(3+1\))-dimensional potential-YTSF equation and a simplified model for reacting mixtures. Chaos Solitons Fractals 34(3), 1006–1013 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yin, H.M., Tian, B., Chai, J., Wu, X.Y., Sun, W.R.: Solitons and bilinear backlund transformations for a (\(3+1\))-dimensional Yu–Toda–Sasa–Fukuyama equation in a liquid or lattice. Appl. Math. Lett. 58, 178–183 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, Y.J., Chen, H.L., Dai, Z.D.: New kink multi-soliton solutions for the (\(3+1\))-dimensional potential-Yu–Toda–Sasa–Fukuyama equation. Appl. Math. Comput. 234, 548–556 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Tu, J.M., Tian, S.F., Xu, M.J., Song, X.Q., Zhang, T.T.: Backlund transformation, infinite conservation laws and periodic wave solutions of a generalized (\(3+1\))-dimensional nonlinear wave in liquid with gas bubbles. Nonlinear Dyn. 83(3), 1199–1215 (2016)

    Article  MATH  Google Scholar 

  20. Xu, M.J., Tian, S.F., Tu, J.M., Ma, P.L., Zhang, T.T.: On quasiperiodic wave solutions and integrability to a generalized (\(2+1\))-dimensional Korteweg-de Vries equation. Nonlinear Dyn. 82(4), 1–19 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Wang, X.B., Tian, S.F., Feng, L.L., Yan, H., Zhang, T.T.: Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (\(3 + 1\))-dimensional variable-coefficient b-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 88(3), 2265–2279 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work is supported by the National Natural Science Foundation of China (Nos. 11675055 and 11435005) and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213).

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Authors and Affiliations

  1. School of Computer Science and Software Engineering, East China Normal University, Shanghai, 200062, China

    Yan Zhang, Yinping Liu & Xiaoyan Tang

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  1. Yan Zhang
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  2. Yinping Liu
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  3. Xiaoyan Tang
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Correspondence to Yinping Liu.

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The authors declare that there are no conflicts of interest between this manuscript and published articles mostly for technical terms, mathematical expressions and explanations on mathematical terms.

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Zhang, Y., Liu, Y. & Tang, X. M-lump and interactive solutions to a (3 \({+}\) 1)-dimensional nonlinear system. Nonlinear Dyn 93, 2533–2541 (2018). https://doi.org/10.1007/s11071-018-4340-9

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  • DOI: https://doi.org/10.1007/s11071-018-4340-9

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