Novel Bäcklund Transformations for Integrable Equations

Pilar Ruiz Gordoa () and Andrew Pickering
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Pilar Ruiz Gordoa: Área de Matemática Aplicada, ESCET Universidad Rey Juan Carlos C, Tulipán s/n, 28933 Móstoles, Madrid, Spain
Andrew Pickering: Área de Matemática Aplicada, ESCET Universidad Rey Juan Carlos C, Tulipán s/n, 28933 Móstoles, Madrid, Spain

Mathematics, 2022, vol. 10, issue 19, 1-9

Abstract: In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation.

Keywords: integrable equations; Bäcklund transformations; Miura maps; inverse matrix; dispersive water wave equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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