Principal Investigator: Eduardo Colorado Heras
The objective of this project is the study of various open problems in the framework of evolution and stationary Partial Differential Equations, motivated by various problems of mathematical physics. The main problems will deal with systems of coupled non-linear Schrödinger type equations, even more, Schrödinger-KdV systems. These kind of systems will study in both local and non-local version, the non-local driven by the fractional Schrödinger operator, since in recent years more and more physicists consider that in order to describe some phenomena of nature in a real way, the Partial Differential Equations with fractional diffusion play a key role because of its non-local character. Although we will be focus on the study of that kind of systems, we will study problems in bounded domains with mixed Dirichlet-Neumann boundary data and in whole the space associated to second order and higher order like for the bi-Laplacian operator. All of them have numerous applications in physics and engineering, from applications in Nonlinear Optics, Plasma, Bose-Einstein Condensates, Tsunami, Population Dinamics, Mathus and Verhulst Laws, study of contaminants in fluids,…but also one has to take into account their mathematical interest. To address the problems we will use techniques of Non-Linear Analysis and Calculus of Variations such as the Theory of critical points (Minimization, Mountain-Pass Theorem, Morse index, etc), Topological Methods (Degree Theory) Bifurcation analysis, etc.