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.

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.

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

• Journal Article Type: Article
• Acceptance Date: Dec 3, 2018
• Online Publication Date: Dec 13, 2018
• Publication Date: Apr 1, 2019
• Deposit Date: Jan 8, 2019
• Publicly Available Date: Jan 8, 2019
• Journal: Physica D: Nonlinear Phenomena
• Print ISSN: 0167-2789
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 391
• Pages: 72-86
• 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
• Public URL: https://nottingham-repository.worktribe.com/output/1450292
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0167278918300812