Bäcklund transformation and applications for the Vakhnenko equation

Min  Xue^a, Hui Mao<sup>b

a</sup> Department of Mathematics, School of Science, China University of Mining and Technology, China
^b School of Mathematics and Statistics, Nanning Normal University, Nanning, Guangxi, China

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.

Keywords: Vakhnenko equation, Bäcklund transformation, nonlinear superposition formula, loop soliton solution.

MSC: 35Q58, 37K10

Received: 12.10.2021
Revised: 18.11.2021

DOI: 10.4213/tmf10182

 English version: Theoretical and Mathematical Physics, 2022, 210:2, 172–183

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