Strong Hyperbolicity of Second-Order PDEs via Matrix Pencils

TL;DR

This paper introduces a new, simplified way to determine strong hyperbolicity of second order PDEs using matrix pencils, showing equivalence to traditional methods and revealing a factorization property.

Contribution

It defines strong hyperbolicity via second order pencils, simplifies hyperbolicity checks, and demonstrates a factorization property for strongly hyperbolic systems.

Findings

Equivalent to standard hyperbolicity definition

Simplifies calculations for checking hyperbolicity

Shows factorization of the second order pencil for strongly hyperbolic systems

Abstract

We introduce a definition of strong hyperbolicity for second order partial differential equations using second order pencils. We show that this definition is equivalent to the standard one, derived by reducing the equations to first order form, but with the benefit of simplifying the calculations necessary to check hyperbolicity. In addition, we observe an interesting property, namely that when a system is strongly hyperbolic, its second order pencil can be factorized as a product of two diagonalizable first order pencils. Finally, we present an application to a vector potential for of Maxwell's equations, with a general extension and gauge fixing.