本项目主要考虑周期介质中时域上波动传播问题的动力学性质。时域上波动现象可获取更多信号信息、模拟更一般介质材料和非线性现象，因此在数学、物理、医学、地学、生命科学、材料科学和信息科学等许多重要科技领域都有广泛的应用。长期以来，时域波动问题动力学行为是理论研究和应用领域高度关注的研究课题之一。. 本项目拟研究时域上周期结构下电磁波和弹性波散射问题解的动力学性质。对双周期介质电磁波散射问题，给出线性Maxwell方程解的适定性和稳定性。考虑非线性电磁波散射问题：给出周期介质中变系数波动方程拟周期解的存在性和稳定性；对双周期介质，给出变系数Maxwell方程周期解的存在性。研究周期结构下弹性波散射问题，给出线性Navier方程解的存在唯一性和稳定性，分析解的渐近性；给出非线性弹性波方程周期解的存在性和正则性。这些研究将为人们深入理解散射现象提供必要的理论依据。

In this project, we will consider the dynamics of scattering problem with periodic structure in the time-domain . The time-domain scattering problems have attracted considerable attention due to their capability of capturing wide-band signals and modeling more general material and nonlinearity. Especially, it has a wide range of applications in mathematical physics, medicine, geosciences, life sciences, materials science , information science and many other important areas of science and technology. Therefore, the dynamics of the scattering problem and the efficient computing method in time-domain have become a great concerned issue both in the fields of theoretical research and practical application for a long time. . The main goal of this proposal is to investigate the existence and stability of quasi-periodic for nonlinear wave equation with variable coefficients and nonlinear Maxwell equation, which corresponding to scattering problem of the periodic dielectric grating and bi-periodic grating, respectively. For the nonlinear Navier equation, we will focuses on the existence and regularity of the periodic solution. In addition, we will study the electromagnetic diffraction of Maxwell's equations by bi-periodic dielectric gratings; the scattering of time-domain plane wave by a elastic medium with periodic medium. We will try to give the well-posedness of the linear scattering problem and the stability of the solution. For the linear Navier equation, we will give the asymptotic property solutions for the elastic wave. We expect that our result will be able to provide some necessary theory basis for people's further understanding of the scattering problem.