Mathematical Sciences: Nonlinear Problems Arising from Theoretical Physics

9400243杨提出了三个主要项目。第一个项目是研究耦合的爱因斯坦和杨-米尔斯-希格斯方程的宇宙弦解。从物理常数和规范群的角度研究耦合的结构、正则驻定解的存在和不存在等问题。这种类型的解在早期宇宙物理和星系形成理论中都有应用。第二个项目是研究由核相互作用的场论框架引起的变分问题。主要问题是在各种同伦类中的所有构型映射中最小化一个能量泛函。同伦类起到了一个具有积分形式的拓扑约束的作用。我们将研究极小化的存在性、规律性和行为。第三个项目是研究Bardeen-Cooper-Schrieffer超导微观理论中的能隙方程。我们将研究临界转变温度的确定、全空间解的唯一性以及超导态的数值近似等问题。这项拟议的研究考虑了理论物理的几个重要领域中出现的非线性问题，包括超导、宇宙学和核系统。数学分析和解决方案将有助于更好地理解这些问题，开辟数学研究的新领域，加强现代数学与物理的互动。所开发的方法也将对其他相关领域的应用、数值和纯数学有所帮助和启发。

9400243 YANG Three main projects are proposed. The first project is a study of the cosmic string solutions of the coupled Einstein and Yang--Mills--Higgs equations. Problems such as the structure of coupling in view of physical constants and gauge groups, existence and nonexistence of regular stationary solutions will be investigated. Solutions of this type have applications in early universe physics and the theory of galaxy formation. The second project is a study of a variational problem arising from a field-theoretical framework of nuclear interactions. The main problem is to minimize an energy functional among all configuration maps in various homotopy classes. The homotopy class plays the role of a topological constraint which takes an integral form. Existence, regularity, and behavior of the minimizers will be investigated. The third project is a study of the energy gap equation in the Bardeen--Cooper--Schrieffer microscopic theory of superconductivity. Problems such as the determination of the critical transition temperature, the uniqueness of a full space solution, and numerical approximations of superconducting states will be investigated. The proposed research considers nonlinear problems arising from several important areas of theoretical physics, including superconductivity, cosmology, and nuclear systems. The mathematical analysis and solutions will help better understand these problems, open new areas of mathematical research, and enhance the interactions of modern mathematics with physics. The methods developed will also be useful and stimulating to other related fields of applied, numerical, and pure mathematics.