Asymptotics for the noncommutative Painlevé II equation

In this paper, we are concerned with the following noncommutative Painlevé II equation {equation*} D^2 β_1 = 4s β_1 +4 β_1 s +8 β_1^3, {equation*} where β₁=β₁(s⃗) is an n × n matrix-valued function of s⃗=(s₁,…,sₙ), s=(s₁,…,sₙ) and D=∑ⱼ₌₁ⁿ∂/∂ sⱼ. If $n=1$, it reduces to the classical Painlevé II equation up to a scaling. Given an arbitrary n × n constant matrix C=(cj k)j, k=1ⁿ, a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution β₁(s⃗)=β₁(s⃗;C) of the noncommutative PII equation such that its $(k,l)$-th entry behaves like -cₖₗ (sₖ+sₗ) as S= 1/n∑ᵢ₌₁ⁿ sⱼ→+∞, where stands for the standard Airy function. For a class of structured matrices C, we establish asymptotics of the associated solutions as S → -∞, 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 → -∞ cannot be deduced solely from its behavior as S → +∞ 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.