Nonlinear Schroedinger systems with saturation effect and Willmore boundary value problem

非线性薛定谔系统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。在研究克尔介质时，人们通常考虑立方非线性薛定谔系统，在过去的十年里，在这一研究领域有许多贡献。在我的研究项目的第一部分，我计划遵循马亚，Montefusco，Pellacci（2013）最近发表的新思想，他们首先系统地分析了具有饱和效应的克尔介质模型。我的目标是，首先，锐化他们的存在结果，在这种材料中的非平凡驻波，其次，处理一类更广泛的非线性薛定谔系统，同样描述饱和非线性material.In我的研究项目的第二部分，我计划处理的曲线和曲面的最小弯曲能量。在数学上，弯曲能量由Willmore能量量化，Willmore能量代表Helfrich能量的简化变体。在细胞生物学中，最小Willmore能量的曲线用作细胞膜的模型。在过去的五年中，人们发现了在所有满足相同边界条件的图形曲线中具有最小Willmore能量的对称图形曲线，在本课题中，我希望证明在所有满足相同边界条件的曲线中具有最优Willmore能量的曲线的存在性。此外，我的目的是延长一些已知的结果对称表面的革命的非对称的情况下，表面的更一般的形状。

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.