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

这项工作分为两个主要主题，非线性焦散中的法向奇点传播和非线性衍射中的同类型奇点传播。这项研究的基础是一个普遍的问题，即微分方程解的过去波前集如何影响未来的波前集。在第一个项目中，将研究二阶严格双曲型方程的柯西问题解在三维空间中的正规奇异性的传播。假设初始数据垂直于具有燕尾奇点的特征超曲面的光滑部分。这项研究的一个目的是要证明，从燕尾点传播的异常奇点最多只有一个锥体。在第二个项目中，将考虑具有Dirichlet条件的半线性双曲混合问题在衍射边界域上的正规奇异性的传播。在这里，目标将是表明，最多有一个锥体的异常奇点传播从掠点。

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.