Mathematical Sciences: "Propagation of Conormal Singularities in Nonlinear Caustics and Nonlinear Diffraction"

这项工作分为两个主要的主题，在非线性焦散中的共法奇点的传播和在非线性衍射中的相同类型的奇点的传播。这项研究的基础是一个一般性的问题，即微分方程解的过去波前集合如何影响未来的波前集合。在第一个项目中，我们将研究三维空间中二阶严格双曲型方程Cauchy问题解的正则奇性的传播。假定初始数据与具有燕尾奇点的特征超曲面的光滑部分正交。这项研究的一个目的是证明至多存在一个从燕尾点传播的异常奇点锥体。在第二个项目中，将考虑具有绕射边界的区域上具有Dirichlet条件的半线性双曲混合问题的法线奇性的传播。这里的目的是证明至多有一个从掠掠点传播的异常奇点的锥体。

This work divides into two main topics, propagation of conormal singularities in nonlinear caustics and the propagation of the same type of singularities in nonlinear diffraction. Underlying this research is the general question of how the past wavefront set of a solution of a differential equation influences the wavefront set in the future. In the first project, work will be done investigating the propagation of conormal singularities of solutions to the Cauchy problem for second order strictly hyperbolic equations in three space dimensions. The initial data is assumed to be conormal to the smooth part of a characteristic hypersurface with a swallowtail singularity. One object of this study will be to show that there is at most a cone of anomalous singularities propagating from the swallowtail point. In the second project, the propagation of conormal singularities for semilinear hyperbolic mixed problems with Dirichlet conditions on domains with diffractive boundaries will be considered. Here, the object will be to show that there is at most a cone of anomalous singularities propagating from the glancing points.