Mónica Clapp: Nonlinear Schrödinger equations with a shrinking region of attraction

Abstract:

We analyze the existence of solutions for a nonlinear Schrödinger equation and for a weakly coupled competitive Schrödinger system in the presence of one or more regions of attraction.

The equation arises, for instance, in the study of optical waveguides propagating through a stratified dielectric medium. The nonlinear term in the equation is related to the contribution of the dielectric response and describes the situation in which the medium exhibits a self-focusing core.

As the regions of attraction shrink, the lower-energy solutions concentrate and blow up. The blow-up profiles satisfy an elliptic equation, or system of equations, with an abrupt change of sign in the nonlinearity. We will describe the concentration behavior and present some qualitative properties of the limit profiles.

These results are the product of collaborations with VÍctor Hernández-SantamarÍa (UNAM), Alberto Saldaña (UNAM) and Andrzej Szulkin (Stockholm University).