In the long-wave regime, nonlinear waves may undergo a phase transition from a smooth to a fast oscillatory behaviour. We show that this phenomenon, commonly known as dispersive shock, shares many features of the tri-critical point in statistical systems and we build a dictionary between nonlinear waves and statistical mechanics. 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.