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High-order lumps, high-order breathers and hybrid solutions for an extended (3 + 1)-dimensional Jimbo–Miwa equation in fluid dynamics

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Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (xy) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (xy) plane are determined by \(k_{12}\) and \(k_{12}p_{11}+k_{11}p_{12}\), while locations are determined by \(k_{11}\) and \(k_{11}p_{11}-k_{12}p_{12}\). Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

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We would like to express our sincere thanks to every member in our discussion group for their valuable comments. The authors would also thank the reviewers for their comments on this paper. The work is supported in part by the National Natural Science Foundation of China under Grant No. 11975145, the Natural Science Foundation of Anhui Province under Grant No. 1408085QA06, the University Excellent Talent Fund of Anhui Province under Grant No. gxyq2019096 and the Natural Science Research Projects in Colleges and Universities of Anhui Province under Grant No. KJ2019A0637.

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Correspondence to Tie-Cheng Xia.

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Guo, H., Xia, T. & Hu, B. High-order lumps, high-order breathers and hybrid solutions for an extended (3 + 1)-dimensional Jimbo–Miwa equation in fluid dynamics. Nonlinear Dyn (2020). https://doi.org/10.1007/s11071-020-05514-9

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  • Hirota bilinear method
  • Extended (3 + 1)-dimensional Jimbo–Miwa equation
  • M-order lumps
  • T-order breathers
  • Hybrid solutions