Asymptotics of Fredholm determinant solutions of the noncommutative Painlev\'e II equation

TL;DR

This paper analyzes the asymptotic behavior of pole-free solutions to the noncommutative Painlevé II equation, expressing results via Fredholm determinants and connecting to classical Painlevé V solutions.

Contribution

It provides the first detailed asymptotic analysis of noncommutative Painlevé II solutions using Riemann-Hilbert methods and relates these solutions to Painlevé V equations.

Findings

Asymptotics of noncommutative Painlevé II solutions derived

Connection formulas for solutions of Painlevé V obtained

Asymptotic behavior characterized in a specific parameter regime

Abstract

In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlev\'e II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the classical Airy operator, which are analogous to the Hastings-McLeod solution and the Ablowitz-Segur solution of the classical Painlev\'e II equation. Using the Riemann-Hilbert approach, we derive the asymptotics of the Fredholm determinant and the associated particular solutions $\beta(\vec{s})$ to the noncommutative Painlev\'e II equation in the regime $\vec{s}=\left(s+\frac{\tau}{\sqrt{-s}},s-\frac{\tau}{\sqrt{-s}}\right)$ with $\tau\ge 0$ and $s\to-\infty$. The solutions depend on a two by two Hermitian matrix with eigenvalues in the interval $(-1,1)$. The asymptotics are expressed in terms of one parameter family of special solutions of the classical Painlev\'e V equation. Furthermore, we derive the asymptotics, including the connection formulas, for this one parameter family of solutions of the Painlev\'e V equation both as $ix\to -\infty$ and $x\to 0$.