Research database

We propose a two–year three–unit research project concerning several mathematical problems arising in the rigorous treatment of nonlinear dispersive equations in presence of singularities. Among dispersive Hamiltonian PDEs, Schrödinger and Dirac equations have becoming cornerstones in the mathematical modelling since their introductions in the XIXth century. The evolution of the experimental techniques in various applied sciences, e.g. condensed and solid matter physics, neurosciences, nanotechnology and so forth, have been gathering through the decades a prominent interest on nonlinear dispersive models. In particular, the last years witnessed a lot of progress in the rigorous analysis of effective nonlinear equations in presence of strong spatial non-homogeneity, inherently calling for the introduction of singularities. Though naturally rising in the applications, such topics triggered a wide horizon of theoretical investigations as well and contributed to fuel flourishing lines of research in pure or applied mathematics. Deep non–homogeneities appear in various fields and in several forms, mainly meant to capture typical complex features of the environment in many natural phenomena. Two common examples are point defects in propagation media and vertices of branching structures and networks. More specifically, the present project focuses on the following two different ways to implement singularities: 1. nonlinear singular equations, where singular potentials, e.g. delta–type potentials, are coupled with nonlinear effective equations; 2. nonlinear equations on singular structures, where standard nonlinear equations are considered on domains as metric graphs or quantum hybrids. Our aim will be twofold. On the one hand, this is a theoretical project primary dedicated to develop the mathematical formalization of such singular models and to push forward the exploration of their phenomenology, which has already proved to be extremely varied. On the other hand, specific attention will be devoted to exploit a rigorous approach to extract relevant insights about features of interest for real world applications.