Hyperbolicity of High Order Systems of Evolution Equations

TL;DR

This paper investigates the hyperbolic properties of high order evolution equations, establishing equivalences between high order and first order hyperbolicity concepts for systems with derivatives up to order four.

Contribution

It defines strong and symmetric hyperbolicity for high order in space evolution equations and proves their equivalence to first order reductions, extending previous concepts.

Findings

Strong hyperbolicity of FTNS systems is equivalent to the existence of a strongly hyperbolic first order reduction.

Symmetric hyperbolicity for FTNS systems (up to N=4) is equivalent to a symmetric hyperbolic first order reduction.

The paper clarifies the relationship between high order and first order hyperbolic systems.

Abstract

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.