双荷Chern-Simons-Higgs涡旋方程组及相关问题研究

Chern-Simons理论是量子场论的一个重要分支,该理论在量子霍尔效应、超导等领域有广泛应用.在该理论中出现了一系列具有丰富数学结构和挑战性的非线性偏微分方程问题.本项目中将对若干Chern-Simons-Higgs方程构组建立双荷(电荷与磁荷)涡旋解的存在性理论.主要包括:(1)对阿贝尔型Chern-Simons-Higgs模型的运动方程及相关问题,建立在全平面以及在紧区域上双荷涡旋解的存在性.(2)对BPS型非阿贝尔Chern-Simons-Higgs涡旋方程组:建立SU(N)(N>6)及一般情形双周期解多解的存在性结果;建立耦合矩阵阶数大于3的情形全平面上非拓扑解的存在性;建立耦合矩阵为一辛矩阵情形时全平面上非径向非拓扑解及双周期区域上解的存在性理论.解决这些问题将为彻底了解Chern-Simons-Higgs模型的双荷涡旋解建立坚实的数学基础,将促进新数学思想方法的产生.

Chern-Simons theory is an important branch of quantum field theory,and it is widely used in the areas such as quantum Hall effects, superconductivity,etc. In this theory there are a series of differential equations which admit colorful mathematical structures and challenges.In this project we will construct dually charged (electric and magnetic charge) vortex solutions for some Chern-Simons-Higgs equations.It mainly contains:(1)Construct dually charged vortex solutions over the full plane and the compact domain for the equations of motion for Abelian Chern-Simons-Higgs model.(2)For BPS type non-Abelian Chern-Simons-Higgs vortex system,establish the multiple existence of doubly periodic solutions for the SU(N)(N>6) and general case;Construct the existence of non-topological solutions over the full plane when the rank of coupling matrix is bigger than 3;Establish the existence of non-radially non-topological solutions and doubly periodic solutions as the coupling matrix is a sympletic one.It will lay a solidly mathematical foundation for fully understanding the dually charged vortex solutions of Chern-Simons-Higgs model and promote the generation of new mathematical ideas by solving these problems.