Mathematical Sciences: Nonlinear Problems Arising from Theoretical Physics

小行星9400243 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。问题，如结构的耦合鉴于物理常数和规范群，存在和不存在的正规固定解决方案将被调查。这种类型的解在早期宇宙物理学和星系形成理论中有应用。第二个项目是研究从核相互作用的场论框架中产生的变分问题。其主要问题是在不同同伦类的所有组态映射中极小化一个能量泛函。同伦类扮演着拓扑约束的角色，它采用积分形式。存在性，规律性和行为的极小将被调查。第三个项目是研究Barnes-库珀-施里弗超导微观理论中的能隙方程。诸如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将被研究。 拟议的研究考虑了理论物理学的几个重要领域，包括超导，宇宙学和核系统所产生的非线性问题。这些数学分析和解决方法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学的相互作用。开发的方法也将是有用的和刺激的应用，数值和纯数学的其他相关领域。

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.