Abstract
With the aid of the reciprocal transformation and the associated Vakhnenko equation, we construct and study a Bäcklund transformation (BT) involving both independent and dependent variables for the Vakhnenko equation. We derive the corresponding nonlinear superposition formula or \(2\)-BT and \(3\)-BT and rewrite them in terms of Pfaffians. We also discuss the representation for the general \(N\)-BT. As applications, some explicit solutions of the Vakhnenko equation are presented.
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Acknowledgments
It is our pleasure to thank Professor Q. P. Liu and the anonymous referees for their useful suggestions and comments.
Funding
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871471, 11931017 and 11905110) and the Natural Science Foundation of Guangxi Zhuang autonomous region, China (Grant No. 2018GXNSFBA050020).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 210, pp. 199-212 https://doi.org/10.4213/tmf10182.
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Xue, M., Mao, H. Bäcklund transformation and applications for the Vakhnenko equation. Theor Math Phys 210, 172–183 (2022). https://doi.org/10.1134/S0040577922020027
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DOI: https://doi.org/10.1134/S0040577922020027