Singularities of hyperbolic systems of partial differential equations.

摘要

Hyperbolic systems of partial differential equations are found often in the study of applied mathematics. A generalization of a technique used to construct multivalued solutions for 2 x 2 first order systems is presented. The generalization is to systems that are not "strictly hyperbolic", i.e., characteristic speeds are not assumed to be unequal. In performing an analysis of stability of this singularity with respect to initial-data, we find the generic singularity type to be {dollar}Sigmasp{lcub}1,1,0{rcub}.{dollar} We also explore several important examples, the n-dimensional Burgers' equation and the Phase-Diffusion Equations of Newell-Cross. For the n-dimensional Burgers' equation, we find that the only stable types of singularities are those that occur canonically (without any constraints on the unfolding map). For the Phase-Diffusion Equations, we use an analysis of leading order terms to show that at the points where the two characteristic speeds are equal, we have a fold, or square-root singularity.