Abstract

The Painlevé II hierarchy is a sequence of nonlinear ODEs, with the Painlevé II equation as first member. Each member of the hierarchy admits a Lax pair in terms of isomonodromic deformations of a rank 2 system of linear ODEs, with polynomial coefficient for the homogeneous case. It was recently proved that the Tracy-Widom formula for the Hastings-McLeod solution of the homogeneous PII equation can be extended to analogue solutions of the homogeneous PII hierarchy using Fredholm determinants of operators acting through higher order Airy kernels. These integral operators are used in the theory of determinantal point processes with applications in statistical mechanics and random matrix theory. From this starting point, this PhD thesis explored the following directions. We found a formula of Tracy-Widom type connecting the Fredholm determinants of operators acting through matrix-valued analogues of the higher order Airy kernels with particular solution of a matrix-valued PII hierarchy. The result is achieved by using a matrix-valued Riemann-Hilbert problem to study these Fredholm determinants and by deriving a block-matrix Lax pair for the relevant hierarchy. We also found another generalization of the Tracy-Widom formula, this time relating the Fredholm determinants of finite-temperature versions of higher order Airy kernels operators to particular solutions of an integro-differential Painlevé II hierarchy. In this setting, a suitable operator-valued Riemann-Hilbert problem is used to study the relevant Fredholm determinant. The study of its solution produces in the end an operator-valued Lax pair that naturally encodes an integro-differential Painlevé II hierarchy. From a more geometrical point of view, we analyzed the Poisson-symplectic structure of the monodromy manifolds associated to a system of linear ODEs with polynomial coefficient, also known as Stokes manifolds. For the rank 2 case, we found explicit log-canonical coordinates for the symplectic 2-form, forming a cluster algebra of specific type. Moreover, the log-canonical coordinates constructed in this way provide a linearization of the Poisson structure on the Stokes manifolds, first introduced by Flaschka and Newell in their pioneering work of 1981.