Hyperbolicity of High Order Systems of Evolution Equations

We study properties of evolution equations which are first order in time and arbitrary order in space (FTNS). Following Gundlach and Mart\'in-Garc\'ia (2006) we define strong and symmetric hyperbolicity for FTNS systems and examine the relationship between these definitions, and the analogous concepts for first order systems. We demonstrate equivalence of the FTNS definition of strong hyperbolicity with the existence of a strongly hyperbolic first order reduction. We also demonstrate equivalence of the FTNS definition, up to N=4, of symmetric hyperbolicity with the existence of a symmetric hyperbolic first order reduction.