场论中偏微分方程的涡旋解

项目名称： 场论中偏微分方程的涡旋解

项目编号： No.11471099

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张瑞凤

作者单位： 河南大学

项目金额： 70万元

中文摘要： 偏微分方程的涡旋解在物理场论中有着广泛的应用和重要的理论意义。 本项目将研究两类来源于物理场论中的涡旋解问题，第一类来源于非线性光学，是由一类连续或离散的非线性Schrodinger型方程（组）控制的光学涡旋；第二类是来源于规范场的非自对偶Chern-Simons涡旋，其数学表现形式为二阶非线性椭圆型方程（组）。 对于第一类问题，我们将综合运用变分方法、不动点方法和加权Sobolev空间等理论，在有界区域或环形区域上研究光学涡旋的存在性；对于第二类问题，我们将应用不定变分方法研究非自对偶Chern-Simons带电涡旋问题，并且通过建立拓扑解、非拓扑解的存在性，分析Chern- - Simons涡旋在无穷远处的衰减速率。 项目的研究将加强数学与数学物理不同分支学科之间的相互交叉和渗透，项目进行中探索的新思想、新方法将促进其它领域出现的涡旋方程（组）的研究，丰富和发展现有的偏微分方程理论。

中文关键词： 涡旋；非线性薛定谔方程；非线性椭圆方程；存在性；非线性分析

英文摘要： It is well known that vortices have broad applications and are of theoretical importance in field theory. In this project, we will study the problems of two classes of vortices arising from field theory in physics. One of them is a class of optical vortices governed by the continuous or discrete nonlinear Schrodinger equations. The other is a class of systems of non-self-dual Chern-Simons vortex equations in gauge field physics, whose mathematical formulation is made in terms of second-order elliptic equations. For the first class of problems, we will establish the existence of optical vortices in bounded disk-like domains and ring-like domains by combining variational methods, fix-point methods, weighted Sobolev spaces and other techniques. For the second class of problems, we will establish the existence of non-self-dual Chern-Simons charged vortices by constrained variational methods applied on indefinite action functionals, and discuss the existence of topological solutions and non-topological solutions and their decay rates at infinity. The new ideas and methods developed in this work will be valuable to other areas of studies concerning vortex equations, enhance the interaction of mathematics and mathematical physics, and enrich the existing theory and methods of partial differential equations.

英文关键词： vortices;nonlinear Schrodinger equations;nonlinear elliptic equations;existence;nonlinear analysis