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Interaction solutions between lump and stripe soliton to the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

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Abstract

In this paper, we apply the ansatz method to the multi-linear form of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation for constructing interaction solutions. By taking the ansatz as the quadratic function or the linear combination of the quadratic function and the exponential one, explicit rational and rational-exponential solutions are derived. It is shown that these exact solutions describe the lump, the lump–stripe soliton interaction with fission and fusion phenomena, and a rogue wave excited from the stripe soliton pair, respectively.

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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, 205–206 (1977)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Gilson, C.R., Nimmo, J.J.C.: Lump solutions of the BKP equation. Phys. Lett. A 147, 472–476 (1990)

    Article  MathSciNet  Google Scholar 

  4. Gaillard, P.: Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves. J. Math. Phys. 57, 063505 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced \(p\)-gKP and \(p\)-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Wang, C.J.: Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation. Nonlinear Dyn. 84, 697–702 (2016)

    Article  MathSciNet  Google Scholar 

  10. Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized (3+1)-dimensional shallow water-like equation. Comput. Math. Appl. 73, 246–252 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, X.E., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24–31 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Zhang, X.E., Chen, Y.: Deformation rogue wave to the (2+1)-dimensional KdV equation. Nonlinear Dyn. 90, 755–763 (2017)

    Article  MathSciNet  Google Scholar 

  14. Wang, Y.H., Wang, H., Dong, H.H., Zhang, H.S., Temuer, C.: Interaction solutions for a reduced extended (3+1)-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 92, 487–497 (2018)

    Article  Google Scholar 

  15. Zhao, H.Q., Ma, W.X.: Mixed lump–kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, J.B., Ma, W.X.: Mixed lump–kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tang, Y.N., Tao, S.Q., Zhou, M.L., Guan, Q.: Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dyn. 89, 429–442 (2017)

    Article  MathSciNet  Google Scholar 

  18. 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 

  19. Peng, W.Q., Tian, S.F., Zou, L., Zhang, T.T.: Characteristics of the solitary waves and lump waves with interaction phenomena in a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation. Nonlinear Dyn. 93, 1841–1851 (2018)

    Article  MATH  Google Scholar 

  20. Chen, M.D., Li, X., Wang, Y., Li, B.: A pair of resonance stripe solitons and lump solutions to a reduced (3+1)-dimensional nonlinear evolution equation. Commun. Theor. Phys. 67, 595–600 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lou, S.Y., Lin, J.: Rogue waves in nonintegrable KdV-type systems. Chin. Phys. Lett. 35, 050202 (2018)

    Article  Google Scholar 

  22. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transormation Groups for Soliton Equations. In: Jimbo, M., Wiwa, T. (eds.) Proceeding of the RIMS Symposium on Nonlinear Integrable Systems-Classical and Quantum Theory. World Scientific, Singapore (1983)

  23. Hu, X.B., Li, Y.: Bäcklund transformation and nonlinear superposition formula of DJKM equation. Acta Math. Sci. 11, 164–172 (1991)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  25. Dorizzi, B., Grammaticos, B., Ramani, A., Winternitz, P.: Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable? J. Math. Phys. 27, 2848–2852 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, Y.H., Wang, H., Temuer, C.: Lax pair, conservation laws, and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation. Nonlinear Dyn. 78, 1101–1107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yuan, Y.Q., Tian, B., Sun, W.R., Chai, J., Liu, L.: Wronskian and Grammian solutions for a (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation. Comput. Math. Appl. 74, 873–879 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11835011 and 11675146.

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  1. Department of Physics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Fan Guo & Ji Lin

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  1. Fan Guo
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  2. Ji Lin
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Correspondence to Ji Lin.

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Guo, F., Lin, J. Interaction solutions between lump and stripe soliton to the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation. Nonlinear Dyn 96, 1233–1241 (2019). https://doi.org/10.1007/s11071-019-04850-9

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  • DOI: https://doi.org/10.1007/s11071-019-04850-9

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