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Instruction on the construction of coherent structures based on variable separation solutions of (2+1)-dimensional nonlinear evolution equations in fluid mechanics

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Abstract

At first, we focus on the projective Riccati equation method again and find that many solutions of the projective Riccati equation in the literatures are not independent. We derive an exponential form solution of the projective Riccati equation, which includes many solutions reported in the literatures. Then, via the projective Riccati equation method with the exponential form solution firstly reported in this paper, we can obtain variable separation solutions of (2+1)-dimensional nonlinear evolution equations in physics and engineering. For example, we get variable separation solutions of (2+1)-dimensional generalized Burgers equation and Boiti–Leon–Pempinelli equation in fluid mechanics. At last, based on these variable separation solutions of two equations, we study coherent structure including the interaction between four-dromion structures and collision between complex wave with dromion pair structures. We find that for the same physical coherent structure of the potential, coherent structures constructed from the original field quantities exhibit physical property (for Burgers equation) and un-physical property (for Boiti–Leon–Pempinelli equation). Therefore, we must be careful with the construction of coherent structures based on variable separation solutions, and we need consider the construction of coherent structures for all field quantities at the same time lest so-called new coherent structures found by researchers which are un-physical and illusive.

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References

  1. Chen, Y.X., Zheng, L.H., Xu, F.Q.: Spatiotemporal vector and scalar solitons of the coupled nonlinear Schroinger equation with spatially modulated cubic-quintic-septimal nonlinearities. Nonlinear Dyn. 93, 2379–2388 (2018)

    Article  Google Scholar 

  2. Wang, Y.Y., Dai, C.Q.: Caution with respect to “new” variable separation solutions and their corresponding localized structures. Appl. Math. Model. 40, 3475–3482 (2016)

    Article  MathSciNet  Google Scholar 

  3. Kong, L.Q., Liu, J., Jin, D.Q., Ding, D.J., Dai, C.Q.: Soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect. Nonlinear Dyn. 87, 83–92 (2017)

    Article  Google Scholar 

  4. Tang, X.Y., Lou, S.Y., Zhang, Y.: Localized exicitations in (2+1)-dimensional systems. Phys. Rev. E 66, 046601 (2002)

    Article  MathSciNet  Google Scholar 

  5. Dai, C.Q., Yan, C.J., Zhang, J.F.: Variable separation solutions in (1+1)-dimensional and (3+1)-dimensional systems via entangled mapping approach. Commun. Theor. Phys. 46, 389–392 (2006)

    Article  MathSciNet  Google Scholar 

  6. Dai, C.Q., Zhou, G.Q.: Exotic interactions between solitons of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. Chin. Phys. 16, 1201–1208 (2007)

    Article  Google Scholar 

  7. Wang, Y.Y., Zhang, Y.P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonlinear Dyn. 83, 1331–1339 (2016)

    Article  MathSciNet  Google Scholar 

  8. Dai, C.Q., Wang, Y.Y.: Localized coherent structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. Nonlinear Dyn. 70, 189–196 (2012)

    Article  MathSciNet  Google Scholar 

  9. Lou, M.R., Zhang, Y.P., Kong, L.Q., Dai, C.Q.: Be careful with the equivalence of different ansatz of improved tanh-function method for nonlinear models. Appl. Math. Lett. 48, 23–29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kong, L.Q., Dai, C.Q.: Some discussions about variable separation of nonlinear models using Riccati equation expansion method. Nonlinear Dyn. 81, 1553–1561 (2015)

    Article  MathSciNet  Google Scholar 

  11. Wang, Y.Y., Dai, C.Q.: Elastic interactions between multi-valued foldons and anti-foldons for the (2+1)-dimensional variable coefficient Broer–Kaup system in water waves. Nonlinear Dyn. 74, 429–438 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ni, W.G., Dai, C.Q.: Note on same result of different ansatz based on extended tanh-function method for nonlinear models. Appl. Math. Comput. 270, 434–440 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Zhang, B., Zhang, X.L., Dai, C.Q.: Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonlinear Dyn. 87, 2385–2393 (2017)

    Article  MathSciNet  Google Scholar 

  14. Dai, C.Q., Wang, Y.Y.: Notes on the equivalence of different variable separation approaches for nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 19, 19–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Emmanuel, Y.: The general projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations. Chin. J. Phys. 43, 991–1003 (2005)

    MathSciNet  Google Scholar 

  16. Huang, D.J., Zhang, H.Q.: Non-travelling wave solutions of the generalized Broer–Kaup system. Chaos Solitons Fractals 23, 601–607 (2005)

    Article  MathSciNet  Google Scholar 

  17. 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, 1006–1013 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dai, C.Q., Zhang, J.F.: Application of He’s exp-function method to the stochastic mKdV equation. Int. J. Nonlinear Sci. Numer. Simul. 10, 675–680 (2009)

    Article  Google Scholar 

  19. Dai, C.Q., Zhang, J.F.: Stochastic exact solutions and two-soliton solution of the Wick-type stochastic KdV equation. Europhys. Lett. 86, 40006 (2009)

    Article  Google Scholar 

  20. Yan, Z.Y.: Singularity structure analysis and abundant new dromion-like structures for the (2+1)-dimensional generalized Burgers equation. Chin. J. Phys. 40, 203–213 (2002)

    MathSciNet  Google Scholar 

  21. Zheng, C.L.: Variable separation approach to solve (2 + 1)-dimensional generalized Burgers system: solitary wave and Jacobi periodic wave excitations. Commun. Theor. Phys. 41, 391–396 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Boiti, M., Leon, J.J.P., Pempinelli, F.: Spectral transform for a two spatial dimension extension of the dispersive long wave equation. Inver. Probl. 3, 371–387 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dai, C.Q., Ni, Y.Z.: Novel interactions between semi-foldons of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. Phys. Scr. 74, 584–590 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dai, C.Q., Wang, Y.Y.: Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. Chaos Solitons Fractals 39, 350–355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lü, Z.S., Zhang, H.Q.: Soliton like and multi-soliton like solutions for the Boiti–Leon–Pempinelli equation. Chaos Solitons Fractals 19, 527–531 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11775104).

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  1. College of Engineering and Design, Lishui University, Lishui, 323000, Zhejiang, China

    Hong-Yu Wu & Li-Hong Jiang

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  1. Hong-Yu Wu
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  2. Li-Hong Jiang
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Correspondence to Hong-Yu Wu.

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Wu, HY., Jiang, LH. Instruction on the construction of coherent structures based on variable separation solutions of (2+1)-dimensional nonlinear evolution equations in fluid mechanics. Nonlinear Dyn 97, 403–412 (2019). https://doi.org/10.1007/s11071-019-04978-8

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