Asymptotics for the noncommutative Painlev\'{e} II equation

TL;DR

This paper studies the asymptotic behavior of solutions to a noncommutative matrix-valued Painlevé II equation, revealing hybrid solution behaviors and connection formulas as the parameter tends to negative infinity.

Contribution

It establishes the asymptotics of solutions for structured matrices in the noncommutative Painlevé II equation, including novel connection formulas and hybrid behaviors.

Findings

Solutions exhibit hybrid behavior combining Hastings-McLeod and Ablowitz-Segur types.

Asymptotics as S→−∞ depend on both entries and their transposes, unlike the scalar case.

Connection formulas relate asymptotics at positive and negative infinity for structured matrices.

Abstract

In this paper, we are concerned with the following noncommutative Painlev\'{e} II equation \begin{equation*} \mathbf{D}^2 \beta_1 = 4\mathbf{s} \beta_1 +4 \beta_1 \mathbf{s} +8 \beta_1^3, \end{equation*} where $\beta_1=\beta_1(\vec{s})$ is an $n \times n$ matrix-valued function of $\vec{s}=(s_1,\ldots,s_n)$, $\mathbf{s}=\diag(s_1,\ldots,s_n)$ and $\mathbf{D}=\sum_{j=1}^n\frac{\partial}{\partial s_j}$. If $n=1$, it reduces to the classical Painlev\'{e} II equation up to a scaling. Given an arbitrary $n \times n$ constant matrix $C=\left(c_{j k}\right)_{j, k=1}^n$, a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution $\beta_1(\vec{s})=\beta_1(\vec{s};C)$ of the noncommutative PII equation such that its $(k,l)$-th entry behaves like $-c_{kl} \Ai (s_k+s_l)$ as $S= \frac{1}{n}\sum_{i=1}^n s_j\to+\infty$, where $\Ai$ stands for the standard Airy function. For a class of structured matrices $C$, we establish asymptotics of the associated solutions as $S \to -\infty$, which particularly include the so-called connection formulas. In the present setting, it comes out that the solution exhibits a hybrid behavior in the sense that each entry corresponds to either an extension of the Hastings-McLeod solution or an extension of the Ablowitz-Segur solution for the PII equation. It is worthwhile to emphasize the asymptotics of the $(k,l)$-th entry as $S \to -\infty$ cannot be deduced solely from its behavior as $S \to +\infty$ in general, which actually also depends on the positive infinity asymptotics of the $(l,k)$-th entry. This new and intriguing phenomenon disappears in the scalar case.