Nonlinear Schroedinger systems with saturation effect and Willmore boundary value problem

具有饱和效应的非线性薛定谔系统和威尔莫尔边值问题

基本信息

批准号：
257041468
负责人：
Dr. Rainer Mandel
金额：
--
依托单位：
Scuola Internazionale Superiore di Studi Avanzati (SISSA)
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2015-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/257041468?language=en
关键词：
Nonlinear Schroedinger systems saturation effect

项目摘要

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.

非线性Schroedinger系统通常用于描述由所谓的非线性材料构建的视波指南中电磁辐射的繁殖。当对Kerr媒体进行研究时，通常会考虑立方非线性Schroedinger系统，在过去的十年中，对该研究领域有很多贡献。在我的研究项目的第一部分中，我计划遵循Maia，Montefusco，Pellacci（2013）最近发表的新想法，他们首先系统地分析具有饱和效果的Kerr媒体模型。我的目的首先是为了加强其在这种材料中的非平凡驻波的结果，其次是处理一类更广泛的非线性Schroedinger系统，这些系统同样描述了饱和的非线性材料。从数学上讲，弯曲能是由代表Helfrich能量的简化变体的Willmore能量来量化的。在最小Willmore能量的细胞生物学曲线中，作为细胞膜的模型。在过去的五年中，在所有满足相同边界条件的图形曲线中，对称图形曲线具有最小的Willmore能量。在我的项目中，我希望证明在满足相同边界条件的所有曲线中具有最佳Willmore能量的曲线。此外，我旨在将革命对称表面的一些已知结果扩展到非对称情况和更一般形状的表面。