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

这项工作分为两个主要议题，传播 非线性焦散线的余法奇点及其传播 在非线性衍射中的同一类型的奇点。 这项研究的基础是一个普遍的问题，即过去是如何 微分方程解的波前集影响 未来的波阵面 在第一个项目中，将调查 Cauchy方程解的余法向奇点的传播 三阶二阶严格双曲型方程的边值问题 空间维度假设初始数据与 特征超曲面的光滑部分 燕尾奇点本研究的一个目的是 表明最多存在一个异常奇点锥 从燕尾点传播。 在第二个项目中，余法线的传播 半线性双曲混合问题的奇性 具有衍射边界的区域上的狄利克雷条件将 被考虑。在这里，目的将是表明，在 大多数是一个异常奇点的圆锥体， 粗略的点。

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.