The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: Scalar and matrix cases
Abstract
In this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation d/dt U_{n} =(2 n  1) ^{(Un+1 Un1)  1}, as well as the scalar case thereof. This equation was recently derived in the context of autoBäcklund 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 componentwise 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.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 April 2019
 DOI:
 10.1016/j.physd.2018.12.001
 Bibcode:
 2019PhyD..391...72P
 Keywords:

 Matrix semidiscrete equations;
 Asymptotic behaviour;
 Hamiltonian formulations of matrix Painlevé equations;
 Solutions of matrix second Painlevé equation;
 Integrable systems