Project Details
Projekt
Nonlinear Schroedinger systems with saturation effect and Willmore boundary value problem
Applicant
Dr. Rainer Mandel
Subject Area
Mathematics
Term
from 2014 to 2016
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 257041468
Nonlinear Schroedinger systems are commonly used to describe the propagataion of electromagnetic radiation in optic wave guides which are built from so-called nonlinear materials. When Kerr media are investigated one usually considers cubic nonlinear Schroedinger systems and during the past ten years there have been many contributions to that research field. In the first part of my research project I plan to follow new ideas that have been recently published by Maia, Montefusco, Pellacci (2013) who were first to systematically analyze a model for Kerr media with saturation effect. My aim is, firstly, to sharpen their existence results for nontrivial standing waves in such materials and secondly, to deal with a broader class of nonlinear Schroedinger systems that equally describe saturated nonlinear materials.In the second part of my research project I plan to deal with curves and surfaces of minimal bending energy. Mathematically the bending energy is quantified by the Willmore energy which represents a simplified variant of the Helfrich energy. In cell biology curves of minimal Willmore energy serve as a model for cell membranes. In the past five years symmetric graph-shaped curves with minimal Willmore energy among all graph-shaped curves satisfying the same boundary conditions have been found. In my project I wish to prove the existence of curves with optimal Willmore energy among all curves which satisfy the same boundary conditions. In addition I aim at extending some known results for symmetric surfaces of revolution to the nonsymmetric case and to surfaces of a more general shape.
DFG Programme
Research Fellowships
International Connection
Italy
Participating Institution
Scuola Internazionale Superiore di Studi Avanzati (SISSA)
