Colloquium: Quantisation effects in dispersive systems.

I will discuss a surprising phenomenon, first observed experimentally in linear optics and quantum wave transmission, and known variously as the Talbot effect, fractalisation, (quantum) revivals, or dispersive quantisation. This phenomenon is manifested in the solution of periodic dispersive equation, starting for a discontinuous initial condition. Then, at times that are rational multiple of the spatial period (“rational” times), the solution is discontinuous, indeed it is built from translated copies of the initial condition, while at all other times, the solution is wildly oscillatory, with positive fractal dimension, but it is continuous, so that the solution has more regularity than the initial state. I will discuss the mathematical description of this phenomenon, and study the effect on it of nonlinearity and of non-periodic boundary conditions, in particular the surprising appearance of logarithmic cusps whenever the initial profile has discontinuities.