The role of discrete group theory in the study of the Nonlinear Schrödinger Equation: from discrete vortices to nonlinear singular optics

 We will present our experience on applying discrete group theory arguments to
study the nonlinear Schrödinger equation. First, we will present the angular
pseudomomentum theory, i.e., a set of results that allows to classify and
predict some properties of the stationary solutions with discrete rotational
symmetry of  such equation. Next, we will use these arguments to study the
dynamical evolution of symmetrical solutions. Finally, we will use all this
knowledge in the study of the dynamical behaviour of phase singularities
of an optical field, a topic that is commonly enclosed in a separated branch of
optics called "nonlinear singular optics".