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.

非线性薛定谔系统通常用于描述电磁辐射在由所谓的非线性材料制成的光波导中的传播。当研究克尔介质时，人们通常会考虑三次非线性薛定谔系统，并且在过去的十年中，人们对该研究领域做出了许多贡献。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了具有饱和效应的 Kerr 介质模型。我的目标首先是提高此类材料中非平凡驻波的存在结果，其次是处理更广泛的非线性薛定谔系统，这些系统同样描述了饱和非线性材料。在我的研究项目的第二部分中，我计划处理具有最小弯曲能量的曲线和曲面。在数学上，弯曲能量由威尔莫尔能量来量化，它代表赫尔弗里希能量的简化变体。在细胞生物学中，最小威尔莫尔能量曲线可用作细胞膜的模型。近五年来，在所有满足相同边界条件的图形曲线中，发现了威尔莫尔能量最小的对称图形曲线。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存在具有最佳威尔莫尔能量的曲线。此外，我的目标是将对称旋转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。