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In some strongly nonlinear regime, solutions of PDEs undergo a phase transition from a regular to a highly-oscillatory behaviour. The transition should be universal and exactly described by some known special functions, called Painleve transcendents. I will introduce this recent field of research focusing on one particular example, the semiclassical limit of the Korteweg-de Vries (KdV) equation. With the help of the audience, I will try to elucidate the similarity between the theory of universality for PDEs and the analogous and celebrated theory for second order phase transitions of statistical mechanics.
