Yang-Mills-Higgs理论中微分方程的磁单极子解

Yang-Mills-Higgs理论描述了丰富的物理现象，为数学研究提供了众多极具挑战性的研究课题。本项目将研究与该理论相关的两类数学问题。第一类是与伸缩子耦合的SU(2)Yang-Mills-Higgs理论中的方程组，第二类是带有Higgs场的SU(2)Yang-Mills-Higgs理论中的方程组。第一类问题的数学结构是带有多个指数项的非线性方程组，将用变分方法和混合射击方法建立该方程组静态球对称解的存在唯一性。第二类问题的数学结构是一类结构复杂的非线性常微分方程组，将用混合射击方法构造其静态球对称整体解，并研究解在边界的渐近行为。对这两类问题的深入研究将不仅有助于理解磁单极子解的意义，而且会发展新的处理非线性微分方程的方法。

Yang-Mills-Higgs theory describes broad ranges of physical phenomena and provides outstanding challenges and opportunities for mathematical research. This proposal aims to study two families of mathematical problems related to the theory. The first family consists of the equations in SU(2) Yang-Mills-Higgs theory coupled to a dilaton. The second family consists of the equations in SU(2) Yang-Mills-Higgs theory with Higgs field. The mathematical structure of the first family is nonlinear equations consist of multiple exponential terms. The existence and uniqueness of the static spherical symmetric solution of the equations will be established by variational methods and multi-shooting methods. The mathematical structure of the second family is a complex structure of nonlinear ordinary differential equations. Multi-shooting methods will be used to construct its global static spherically symmetric solution and the asymptotic behavior of the solution at the boundary will be studied. This in-depth research will not only help understand the meaning of the monopole solutions but also develop a series of new methods for the study of nonlinear differential equations of important scientific value.