  The aim of this paper is threefold. First we display solutions of the cubic
nonlinear Schr{\"o}dinger equation on R in link with initial data a sum of
Dirac masses. Secondly we show a Talbot effect for the same equation. Finally
we prove the existence of a unique solution of the binormal flow with datum a
polygonal line. This equation is used as a model for the vortex filaments
dynamics in 3-D fluids and superfluids. We also construct solutions of the
binormal flow that present an intermittency phenomena. Finally, the solution we
construct for the binormal flow is continued for negative times, yielding a
geometric way to approach the continuation after blow-up for the 1-D cubic
nonlinear Schr{\"o}dinger equation.

Analysis of PDEs