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Bäcklund transformation and N-shock-wave solutions for a (3+1)-dimensional nonlinear evolution equation

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Abstract

In this paper, a (3+1)-dimensional nonlinear evolution equation is investigated, which can be used to describe reacting mixtures and shallow water waves. Through the Hirota method and symbolic computation, bilinear forms and Bäcklund transformation are derived, which are different from those in the existing literature. Moreover, N-shock-wave solutions are obtained. Based on those shock-wave solutions, propagation and collision of the shock waves are discussed via the asymptotic and graphic analysis on different planes: (1) oblique elastic collisions between/among the two/three shock waves will arise on the x–y and y–z planes, while parallel elastic collisions exist on the x–z plane; (2) shock waves maintain their original directions, amplitudes and velocities except for some small phase shifts after each collision; (3) the shock wave with higher amplitude travels faster and moves across the slower.

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Notes

  1. Oblique collision means that the angle of the collision between two shock waves is nonzero [36].

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Acknowledgments

This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Ya Sun, Bo Tian, Yu-Feng Wang & Hui-Ling Zhen

  2. State Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Bo Tian

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  1. Ya Sun
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Correspondence to Ya Sun.

Appendix: Some exchange formulas of the Hirota bilinear operator

Appendix: Some exchange formulas of the Hirota bilinear operator

The following exchange formulas of the Hirota bilinear operator hold for the real arbitrary functions f and \(f'\) [28–30]:

$$\begin{aligned}&(D_{y}^{2}f'\cdot f')f^{2}-(D_{y}^{2}f\cdot f)f'^{2}=2D_{y}(D_{y}f'\cdot f)\cdot (f'f),\\&D_{x}(D_{z}f'\cdot f)\cdot (f'f)=D_{z}(D_{x}f'\cdot f)\cdot (f'f),\\&D_{x}^{3}(D_{z}f'\cdot f)\cdot (f'f) \\&\quad =D_{z}(D_{x}^{3}f'\cdot f)\cdot (f'f)-3D_{z}(D_{x}^{2}f'\cdot f)\cdot (D_{x}f'\cdot f),\\&(D_{z}D_{t}f'\cdot f')f^{2}-(D_{z}D_{t}f\cdot f)f'^{2} \\&\quad =2D_{z}(D_{t}f'\cdot f)\cdot (f'f),\\&4(D_{x}^{3}D_{z}f'\cdot f')f^{2}-4(D_{x}^{3}D_{z}f\cdot f)f'^{2} \\&\quad =2D_{z}(D_{x}^{3}f'\cdot f)\cdot (f'f)+6D_{x}(D_{x}^{2}D_{z}f'\cdot f)\cdot (f'f)\\&\qquad -\,6D_{x}(D_{x}^{2}f'\cdot f)\cdot (D_{z}f'\cdot f){-}\,6D_{z}(D_{x}^{2}f'\cdot f)\cdot (D_{x}f'\cdot f) \\&\qquad -\,12D_{x}(D_{x}D_{z}f'\cdot f)\cdot (D_{x}f'\cdot f),\\&4D_{z}(D_{x}^{2}f'\cdot f)\cdot (D_{x}f'\cdot f)+D_{x}^{3}(D_{z}f'\cdot f)\cdot (f'f) \\&\quad =D_{x}[(D_{x}^{2}D_{z}f'\cdot f)-(D_{x}^{2}f'\cdot f)\cdot (D_{z}f'\cdot f) \\&\qquad +\,2(D_{x}D_{z}f'\cdot f)\cdot (D_{x}f'\cdot f)]. \end{aligned}$$

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Sun, Y., Tian, B., Wang, YF. et al. Bäcklund transformation and N-shock-wave solutions for a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn 84, 851–861 (2016). https://doi.org/10.1007/s11071-015-2531-1

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