On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems obtained as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the full equations are approximately described by particular solutions of the Painlev'e-I hierarchy. As concrete examples we discuss nonlinear Schrodinger equations in the semiclassical limit. The conjecture is extended to include blow-up in the semiclassical limit. A numerical study of concrete examples provides strong evidence in support of the conjecture