Abstract
We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.
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Acknowledgements
The authors express their sincere thanks to the referees for their valuable comments. This work was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20181351), the ‘Qinglan Engineering project’ of Jiangsu Universities, the National Natural Science Foundation of China (Grant No. 11301527), the Fundamental Research Fund for the Central Universities (Grant No. 2019QNA35), and the General Financial Grant from the China Postdoctoral Science Foundation (Grant Nos. 2015M570498, 2017T100413).
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Wang, H., Tian, S., Zhang, T. et al. Lump wave and hybrid solutions of a generalized (3 + 1)-dimensional nonlinear wave equation in liquid with gas bubbles. Front. Math. China 14, 631–643 (2019). https://doi.org/10.1007/s11464-019-0775-7
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DOI: https://doi.org/10.1007/s11464-019-0775-7
Keywords
- Generalized (3 + 1)-dimensional nonlinear wave equation
- bilinear formalism
- soliton solutions
- lump solutions
- hybrid solutions