Gordoa, Pilar RPickering, Andrew2024-12-022024-12-022022Gordoa, P.R.; Pickering, A. Novel Bäcklund Transformations for Integrable Equations. Mathematics 2022, 10, 3565. https://doi.org/10.3390/math101935652227-7390https://hdl.handle.net/10115/42255The research in this paper was funded by the Ministry of Science and Innovation of Spain under contract PID2020-115273GB-100 (AEI/FEDER, EU).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.engAtribución 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/integrable equationsBäcklund transformationsMiura mapsinverse matrix dispersive water wave equationinfo:eu-repo/semantics/article10.3390/math10193565info:eu-repo/semantics/openAccess