Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.Comment: 19 page

Applications of Temperley-Lieb algebras to Lorentz lattice gases

Motived by the study of motion in a random environment we introduce and investigate a variant of the Temperley-Lieb algebra. This algebra is very rich, providing us three classes of solutions of the Yang-Baxter equation. This allows us to establish a theoretical framework to study the diffusive behaviour of a Lorentz Lattice gas. Exact results for the geometrical scaling behaviour of closed paths are also presented.Comment: 10 pages, latex file, one figure(by request

On an isomonodromy deformation equation without the Painlev\ue9 property

We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation PI2 compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2 7 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlev\ue9 property. We also study the properties of the Riemann-Hilbert problem associated to this ODE and find its large-t asymptotic solution for physically interesting initial data. \ua9 2014 Pleiades Publishing, Ltd