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

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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.