Asymptotics for the noncommutative Painlevé II equation

Abstract

In this paper, we are concerned with the following noncommutative Painlevé II equation \begin{equation*} \mathbf{D}^2 β_1 = 4\mathbf{s} β_1 +4 β_1 \mathbf{s} +8 β_1^3, \end{equation*} where ${\hat{I}}_1^2={\hat{I}}_1^2(\vec{s})$ is an $n\times n$ matrix-valued function of $\vec{s}=(s_1,\dots ,s_n)$, $\mathbf{s}=\diag(s_1,\ldots,s_n)$ and $\mathbf{D}={\sum }_{j=1}^n\frac{\partial }{\partial s_j}$. If $n=1$, it reduces to the classical Painlevé II equation up to a scaling. Given an arbitrary $n\times n$ constant matrix $C={\left(c_{jk}\right)}_{j,k=1}^n$, a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution ${\hat{I}}_1^2(\vec{s})={\hat{I}}_1^2(\vec{s};C)$ 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.