Critical behaviour for scalar nonlinear waves

In the long-wave regime, nonlinear waves may undergo a phase transition from a smooth to a fast oscillatory behaviour. We study this phenomenon, commonly known as dispersive shock, in the light of Dubrovin's universality conjecture , and we argue that the transition can be described by a special solution of a model universal partial differential equation. This universal solution is constructed by means of a string equation. We provide a classification of universality classes and the explicit description of the transition by means of special functions, extending Dubrovin's universality conjecture to a wider class of equations. In particular, we show that Benjamin-Ono equation belongs to a novel universality class with respect to the ones known in the literature, and we compute its string equation exactly. We describe our results using the language of statistical mechanics, showing that dispersive shocks share many features of the tri-critical point in statistical systems, and building a dictionary between nonlinear waves and statistical mechanics.