Higher-order mixed localized wave solutions and bilinear auto-Bäcklund transformations for the (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Abstract

In this article, the complex conjugate condition technique and the long wave limit method are proposed to obtain higher-order mixed localized wave solutions of the (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation. Three types of bilinear auto-Bäcklund transformations are derived via the Hirota bilinear method. In order to provide rich localized structures, the N-soliton solutions are supplemented via numerical simulation, which produces the hybrid solution of bell-shaped waves, periodic-breather waves and lump waves. The dynamical behaviors of mixed localized wave structures are demonstrated graphically via three-dimensional profiles for suitable values of the arbitrary free parameters. Meanwhile, different structures of the N-soliton solutions are systematically obtained, including different combinations of a-order bell-shaped waves, b-order periodic-breather waves and c-order lump waves. The proposed method can be better to study the novel localized wave solutions of nonlinear evolution equations, the results obtained can provide useful information for analyzing the theory of fluid mechanics, ocean dynamics and plasma physics.

Appendix

$$\begin{aligned} \begin{aligned} \beta _{ij}&=\frac{12\lambda _{1}+6\lambda _{5}(m_{i}+m_{j})}{\lambda _{4}(m_{i}-m_{j})^{2}}, \theta _{i}=-x-m_{i}y-p_{i}z+(\lambda _{4}m_{i}^{2}+\lambda _{9}p_{i})t, B_{ijk}=B_{ij}B_{ik}B_{jk}, \\ \delta _{ij}&=\frac{6k_{j}[2\lambda _{1}+5\lambda _{3}k_{j}^{2}+\lambda _{5}(m_{i}+m_{j})]}{k_{j}^{2}[3\lambda _{1}+5\lambda _{3}k_{j}^{2}+\lambda _{5}(m_{i}+2m_{j})]-\lambda _{4}(m_{i}-m_{j})^{2}}, B_{ijkm}=B_{ij}B_{ik}B_{im}B_{jk}B_{jm}B_{km}. \end{aligned} \end{aligned}$$

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