Pickering, AndrewGordoa, Pilar RWattis, Jonathan A D2024-12-022024-12-022019Physica D 391 (2019) 72–86Andrew Pickering, Pilar R. Gordoa, Jonathan A.D. Wattis, The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: Scalar and matrix cases, Physica D: Nonlinear Phenomena, Volume 391, 2019, Pages 72-86, ISSN 0167-2789, https://doi.org/10.1016/j.physd.2018.12.0010167-2789https://hdl.handle.net/10115/42249We are grateful to the Ministry of Economy and Competitiveness of Spain for funding under grant number MTM2016-80276-P (AEI/FEDER, EU).In this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation U_{n,t} = (2n − 1)(U_{n+1} − U_{n−1})^{-1}, as well as the scalar case thereof. This equation was recently derived in the context of auto-Bäcklund transformations for a matrix partial differential equation. We use asymptotic techniques to reveal a connection between this equation and the matrix (or, as appropriate, scalar) first Painlevé equation. In the matrix case, we also discuss our asymptotic analysis more generally, as well as considering a component-wise approach. In addition, Hamiltonian formulations of the matrix first and second Painlevé equations are given, as well as a discussion of classes of solutions of the matrix second Painlevé equation.engAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/matrix semidiscrete equationsasymptotic behaviourHamiltonian formulations of matrix Painlevé equationssolutions of matrix second Painlevé equationintegrable systemsinfo:eu-repo/semantics/article10.1016/j.physd.2018.12.001info:eu-repo/semantics/openAccess