Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions

We begin with presentation of classification results in the theory of Hamiltonian PDEs with one spatial dimension depending on a small parameter. Special attention is paid to the deformation theory of integrable hierarchies, including an important subclass of the so-called integrable hierarchies of the topological type associated with semisimple Frobenius manifolds. Many well known equations of mathematical physics, such as KdV, NLS, Toda, Boussinesq etc., belong to this subclass, but there are many new integrable PDEs, some of them being of interest for applications. Connections with the theory of Gromov-Witten invariants and random matrices are outlined. We then address the problem of comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their Hamiltonian perturbations containing higher derivatives. We formulate Universality Conjectures describing different types of critical behavior of perturbed solutions near the point of gradient catastrophe of the unperturbed one.