Phase transitions in solutions of nonlinear PDEs and in random matrices

The talk is devoted to study of solutions of nonlinear evolutionary PDEs with slowly changing initial conditions. For the solutions of such equations various types of phase transitions are observed; the qualitative type of solutions' behaviour is changed at the points of phase transitions. In particular, solutions of Hamiltoninan PDEs can have zones of fast oscillations. We will formulate a universality conjecture on asymptotic description of solutions at the points of phase transitions. The motivation of this conjecture will be presented, and some rigorous results will be formulated. Much attention will be given to nonlinear PDEs that originate from the theory of random matrices.