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

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 $x\to 0$.