  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.
