Nonlinear Schrödinger equations with point interactions

Abstract

Interest in the study of non-linear Schrodinger equations with points of interaction has been growing rapidly in recent years by their deep applications in physics, non-linear optics and networks. From a mathematical point of view, the problems that arise are generally non-standard and implying the creation of new strategies and tools in their study. These models have been generally studies on the line or in the periodic case, such as has been done by the proponents in recent years. The case of studying these non-linear models on graphs (star graph) has attracted a lot of attention in the last 5 years, either because of their implications in nano-technology and networks, or because it is becoming a new field of mathematical research. In this research project we are interested with Prof. Ohta in developing new techniques for the study of “standing-wave” solutions for new models on star graphs. Our immediate objectives in this project related to Schrodinger-type models will be the study of the possibility of blow-up solutions for interaction models on the line and the study of non-linear stability of standing-wave with “bump” profiles for models on star graph. Another model that has called our attention of research is the Schrodinger-KdV type system (by our past studies on this models), that arise in the study of the interaction of water waves of type long-wave and short-wave. In this case our interest is to use variational methods to study the existence and stability (instability) of new solutions of solitary-wave type. (AU)