夸克禁闭机制中出现的偏微分方程问题

项目名称： 夸克禁闭机制中出现的偏微分方程问题

项目编号： No.11471100

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： 杨亦松

作者单位： 河南大学

项目金额： 75万元

中文摘要： 夸克禁禁闭机制是物理学中的一大难题。一个广为接受的禁闭机制是线性禁闭。过去的十年中，在超对称量子场论中出现了大量结构新颖、富有兴趣和挑战性的非线性偏微分方程问题。尽管场理论取得了很大成功，但是对这些问题的严格数学结果却并不多见。本项目中申请人将研究两大类出现于夸克禁闭机制中的非线性偏微分方程问题。第一类问题是关于统一非阿贝尔磁单极子和涡旋的相互作用的非线性方程组。我们将构造这类方程组的解并得到所期望的线性禁闭机制。第二类问题主要研究带色荷的(非阿贝尔)涡旋的控制方程组，这对对偶Meissner效应的实现是基本的，也是研究线性禁闭机制的一个起点。对这些方程组的深入研究不仅为夸克禁闭机制本身提供坚实的基础，而且为最终实现磁单极子、涡旋统一方程组解的构造铺平了道路。项目中发展的新思想和方法对其它领域出现的涡旋方程组也是有价值的，同时将加强数学和物理的相互促进，丰富和发展现有的偏微分方程理论。

中文关键词： 规范场方程；夸克禁闭；孤立子；非线性椭圆方程；非线性分析

英文摘要： A well-known great puzzle in modern physics is quark confinement. A widely accepted quark confinement mechanism is linear confinement. In this important area of scientific research, many challenging and interesting nonlinear partial differential equation problems with novel structures and of fundamental nature have been derived over the last ten years by physicists in the context of supersymmetric quantum field theory. However, despite the success of the field-theoretical formalism, few mathematically rigorous investigations regarding these nonlinear partial differential equations are available. In this project the principal investigator will study two main classes of nonlinear partial differential equation problems arising in quak confinement mechanism in supersymmetric quantum field theory described above. The first class of the problems will be about some systems of nonlinear partial differential equations governing the unified interaction of non-Abelian monopoles, modeling quarks, and vortexlines, modeling the forcelines which bond the quarks together. We shall aim at constructing the solutions of these systems of nonlinear equations and obtaining the anticipated linear confinement scenario. The second class of the problems will be about some systems of nonlinear equations governing color-charged (non-Abelian) vortexlines which are essential for the realization of the dual Meissner effect and serve as the very starting point for the linear confinement mechanism. An in-depth understanding of these systems of equations prepares a solid foundation for the quark confinement mechanism itself and paves the path to an ultimate solution of the unified monopole-vortex equation problems. 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 physics, and enrich the existing theory and methods of partial differential equations.

英文关键词： Gauge field equations;quark confinement;solitons;nonlinear elliptic equations;nonlinear analysis