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The higher-order lump, breather and hybrid solutions for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in fluid mechanics

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Under investigation in this paper is the (2 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation, which can be utilized to describe certain nonlinear phenomena in fluid mechanics. We obtain the higher-order lump, breather and hybrid solutions, and analyze the effects of the constant coefficients \(h_1\), \(h_2\), \(h_4\) and \(h_5\) in that equation on those solutions, since the higher-order lump solutions are generalized via the long-wave limit method, and since the higher-order breather solutions and hybrid solutions composed of the solitons, breathers and lumps are derived. With the help of the analytic and graphic analysis, we get the following: (1) amplitudes of the humps and valleys of the first-order lumps are related to \(h_1\), \(h_2\), \(h_4\) and \(h_5\), proportional to \(h_4\) while inversely proportional to \(h_2\). Velocities of the first-order lumps are proportional to \(h_4\). The second-order lumps describe the interaction between the two first-order lumps, which is elastic since those lumps keep their shapes, velocities and amplitudes unchanged after the interaction. Effects of \(h_2\) and \(h_4\) on the second-order lumps are graphically illustrated. (2) Amplitudes of the first-order breathers are proportional to \(h_2\). Interaction between the breather waves is graphically presented. Effects of \(h_2\) and \(h_1\) on the amplitudes and shapes of the second-order breathers are graphically discussed. (3) Elastic interactions are graphically illustrated, between the first-order breathers and one solitons, the first-order lumps and one solitons, as well as the first-order breathers and first-order lumps. Also graphically illustrated, amplitudes of all those three kinds of hybrid solutions are inversely proportional to \(h_2\), and velocity of the one soliton is positively correlated to \(h_4\).

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Acknowledgements

We express our sincere thanks to the editors, reviewers and members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023.

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  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Cai-Yin Zhang, Yi-Tian Gao, Liu-Qing Li & Cui-Cui Ding

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Zhang, CY., Gao, YT., Li, LQ. et al. The higher-order lump, breather and hybrid solutions for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in fluid mechanics. Nonlinear Dyn 102, 1773–1786 (2020). https://doi.org/10.1007/s11071-020-05975-y

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