Rational solutions of nonautonomous quadrilateral equations by the bilinearization of Bäcklund transformation systems

Danda Zhang^a, Liya Zhu^a, Yingying  Sun<sup>b

a</sup> School of Mathematics and Statistics, Ningbo University, Ningbo, China
^b Department of Mathematics, University of Shanghai for Science and Technology, Shanghai, China

Abstract: Rational solutions of several nonautonomous quadrilateral equations in the ABS and ABS* list are obtained in a neat form of Casoratians, which mostly relies on a single $\tau$ function. The corresponding nonautonomous bilinear equations are listed in difference and differential–difference forms by introducing an auxiliary variable. Instead of bilinearizing quadrilateral equations, we present their related Bäcklund transformation systems, which directly reduce to bilinear equations by specific transformations. As an application, a result related to the discrete Painlevé equation is given.

Keywords: rational solutions, nonautonomous, Bäcklund transformation, quadrilateral equations.

Received: 12.09.2024
Revised: 22.01.2025

DOI: 10.4213/tmf10826

 English version: Theoretical and Mathematical Physics, 2025, 223:1, 576–596

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