Critical behaviour in nonlinear Hamiltonian PDEs and Painlevé transcendents

We will consider phase transitions from regular to oscillatory behaviour of solutions to nonlinear Hamiltonian PDEs. Such phenomena were first observed for solutions to the Korteweg–de Vries (KdV) equation. The problem of such a critical behaviour in more general nonintegrable PDEs and systems of PDEs remains essentially unexplored. We propose simple arguments, partially supported by rigorous results as well as numerical evidences, that even in the general case a kind of local integrability holds at the point of phase transition. This gives a possibility to obtain an asymptotic description of the critical behaviour in terms of certain particular solutions of the Painlevé equations and their generalizations.