D’Alembert wave and soliton molecule of the modified Nizhnik–Novikov–Veselov equation

Abstract

The wave motion equation is one of the fundamental equations to describe vibrations of continuous systems. The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential equations. The study of the D’Alembert wave deserves deep consideration in nonlinear equations. In this paper, the D’Alembert-type wave of the (2 + 1)-dimensional modified Nizhnik–Novikov–Veselov (mNNV) equation is derived by the Ansätze method. The Hirota bilinear form of the mNNV equation is constructed by introducing the dependent variable transformation. The multi-soliton solution is obtained by solving the corresponding bilinear form. By combining the velocity resonance mechanism, a three-soliton molecule, the interaction between a soliton molecule and one soliton, and the interaction between a soliton–solitoff molecule and one soliton of the mNNV equation are obtained. The dynamics of these solutions are shown by selecting the appropriate parameters. These phenomena for the mNNV equation have not yet been given via other methods.

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Data availability statement

This manuscript has associated data in a data repository. [Author’s comment: All data included in this manuscript are available upon request by contacting with the corresponding author.]

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Acknowledgements

This work is supported by the National Natural Science Foundation of China Nos. 11775146 and 11835011. The authors are indebted to thank S.Y. Lou for useful discussions.

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Correspondence to Bo Ren or Ji Lin.

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Ren, B., Lin, J. D’Alembert wave and soliton molecule of the modified Nizhnik–Novikov–Veselov equation. Eur. Phys. J. Plus 136, 123 (2021). https://doi.org/10.1140/epjp/s13360-021-01099-3

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