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Painlevé analysis, auto-Bäcklund transformations, bilinear form and analytic solutions on some nonzero backgrounds for a \((2{+}1)\)-dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics

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Abstract

Fluid mechanics concerns the mechanisms of liquids, gases and plasmas and the forces on them. We aim to investigate a \((2+1)\)-dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics in this paper. With the help of the Painlevé analysis, we find that the above system has Painlevé-integrable property. A set of the auto-Bäcklund transformations and some solutions are derived by the virtue of the truncated Painlevé method. We obtain certain bilinear forms via some seed solutions. According to the mentioned bilinear form, we derive the multiple-soliton solutions on some nonzero backgrounds. Based on the soliton solutions and conjugation transformations, the higher-order breather solutions on certain nonzero backgrounds have been obtained. Via some conjugation transformations, hybrid solutions formed from the breathers and solitons on certain nonzero backgrounds have been derived. We also graphically show the interactions between those solitons and breathers.

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Notes

  1. The bilinear form in this paper is different from those in Refs. [11, 12].

  2. This symbol calculation method based on the neural networks may present a general symbolic computing path for the analytic solutions of the nonlinear partial differential equations [24,25,26,27, 48,49,50,51].

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by BUPT Excellent Ph.D. Students Foundation under Grant No. CX2023302.

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BUPT Excellent Ph.D. Students Foundation under Grant No. CX2023302.

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

    Tian-Yu Zhou, Bo Tian, Yuan Shen & Chong-Dong Cheng

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  1. Tian-Yu Zhou
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  2. Bo Tian
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Zhou, TY., Tian, B., Shen, Y. et al. Painlevé analysis, auto-Bäcklund transformations, bilinear form and analytic solutions on some nonzero backgrounds for a \((2{+}1)\)-dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09450-w

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