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The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: scalar and matrix cases

Pickering, Andrew; Gordoa, Pilar R.; Wattis, Jonathan A.D.

Authors

Andrew Pickering

Pilar R. Gordoa



Abstract

In this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation D dtUn = (2n − 1)(Un+1 − Un−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.

Journal Article Type Article
Publication Date Apr 1, 2019
Journal Physica D: Nonlinear Phenomena
Print ISSN 0167-2789
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 391
Pages 72-86
APA6 Citation Pickering, A., Gordoa, P. R., & Wattis, J. A. (2019). 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, 391, 72-86. https://doi.org/10.1016/j.physd.2018.12.001
DOI https://doi.org/10.1016/j.physd.2018.12.001
Keywords Matrix semidiscrete equations; Asymptotic behaviour; Hamiltonian formulations of matrix Painlevé equations; Solutions of matrix second Painlevé equation; Integrable systems
Publisher URL https://www.sciencedirect.com/science/article/pii/S0167278918300812

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