Classical solutions of inviscid Burgers' equation blow-up in a finite time, as they develop a singularity called 'gradient catastrophe'.

We address the problem of studying the 'dispersive shock' phenomenon - how the inviscid Burgers' equation is regularized by a Hamiltonian (also called dispersive or conservative) perturbation, such as the Korteweg-de Vries equation. According to a conjecture due to Dubrovin, the regularisation of the gradient catastrophe is universal - independent of the initial data and of the Hamiltonian perturbation.

We present some preliminary results towards the proof of the conjecture.

This is a joint work with A. Raimondo, SISSA, Trieste.