Schrodinger equations with point interactions and instability for the fractional Korteweg- de Vries equation

Abstract

The aim of this research project is the qualitative study of Schrodinger type models with points interaction determined by the delta distribution of Dirac or its derivative, as well as the study of the fractional Korteweg-de Vries equation. These research topics fit within the area of non-linear dispersive equations. Our research will be directed to the existence and stability or instability of traveling waves solutions and to the Cauchy problem, among others.These specific models are of recent research from a physical point of view in non-linear optics, propagation of optical beams in non-canonical means and graphs (for the Schrodinger type models) and water wave theory (for models type Korteweg-de Vries). The proposed research topic and the specific models being considered, in general, has been little development from the mathematical point of view. Several recent results in this topic have been obtained by the applicant and its collaborators as well as the associated scholarship supervisor that is being claimed (Prof. J-C Saut). In the case of Schrodinger type models with points of interaction, the applicant has used the extension theory for symmetric operators of Krein & von Neumann to simplify and extend recent results. For the well-posedness Cauchy problem associated with these models, we are interested in obtaining results of existence of solutions, uniqueness and continuous dependence of the solutions with respect to the initial data. (AU)