  We review various methods for the analysis of initial-value problems for
integrable dispersive equations in the weak-dispersion or semiclassical regime.
Some methods are sufficiently powerful to rigorously explain the generation of
modulated wavetrains, so-called dispersive shock waves, as the result of shock
formation in a limiting dispersionless system. They also provide a detailed
description of the solution near caustic curves that delimit dispersive shock
waves, revealing fascinating universal wave patterns.
