Talbot effect for the cubic nonlinear Schr\"odinger equation on the torus

We study the evolution of the one dimensional periodic cubic Schr\"odinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was first observed by Talbot in an optical experiment performed in 1836. In this paper we prove that a similar phenomenon occurs in the case of the NLS equation.