非线性Dirac方程和Dirac系统在量子物理中用来描述费米子的运动状态.Dirac-调和映射是量子力学中超对称非线性σ-模型的几何分析版本，也是黎曼几何中调和映射的自然推广，Dirac-测地线是1维情形的Dirac-调和映射. 本项目首先利用变分方法研究紧spin流形上带有位势项的非线性Dirac方程解的存在性、多重性、解的性态,以及具有临界增长的非线性Dirac方程和Dirac系统解的存在性和多重性；其次，拟结合热流方法和Morse理论，在目标流形满足适当条件下，证明Dirac-测地线的存在性和多重性；最后，拟采用变分方法，证明n维球面上非线性Dirac-调和映射的存在性.

Nonlinear Dirac equations and Dirac systems are used to describe the motion of the fermions in quantum physics. Dirac-Harmonic maps are a geometric analytic model corresponding to the supersymmetric nonlinear σ-model in quantum mechanics and a natural extension of the harmonic maps in Riemannian geometry. When the dimension of the domain manifold is one, Dirac - harmonic maps are called Dirac - geodesics . In this projection, firstly, the variational methods are used to study the existence, multiple solutions and the properties of solutions of nonlinear Dirac equation with potential terms on the compact spin manifold, as well as the nonlinear Dirac equation and the Dirac system with critical growth; secondly, by combining the heat flow method and the Morse theory, the existence and multiple solutions of the Dirac- geodesic are proved under the certain conditions of the target manifolds; finally, the existence of nonlinear Dirac- harmonic maps on the n-dimensional sphere are proved by the variational methods.