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Strong Hyperbolicity of Second-Order PDEs via Matrix Pencils

Fernando Abalos, David Hilditch

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.

Strong Hyperbolicity of Second-Order PDEs via Matrix Pencils

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.
Paper Structure (19 sections, 8 theorems, 152 equations, 1 figure)

This paper contains 19 sections, 8 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strong hyperbolicity at first order in time derivatives
  3. First and second order pencils
  4. Pencil Diagonalization
  5. Diagonalization of $M\left(\lambda\right)$
  6. $\mathbf{S}(\lambda)$ as a product of two first order pencils
  7. The $B=0$ case
  8. Uniformity condition
  9. Strong hyperbolicity at second order in time derivatives
  10. Applications
  11. Almost wave equation in 1+1
  12. Electrodynamics
  13. Extension
  14. Gauge fixing
  15. Strong hyperbolicity
  16. ...and 4 more sections

Key Result

Lemma 1

For any $\lambda$,

Figures (1)

  • Figure 1: In these figures, we present the four different cases (\ref{['caso_1']}), (\ref{['caso_2']}), (\ref{['caso_4']}), and (\ref{['caso_6']}). The plots depict some "horizontal" projection of the null cones associated with the metrics: $\textcolor{navy}{g_{ab}}$ (blue), $\textcolor{teal}{\tilde{g}_{ab}}$ (green), and $\textcolor{DarkRed}{\hat{g}_{ab}}$ (red). In case (1), the null cones do not intersect; in case (2), there is a single intersection between the null cones of $\textcolor{navy}{g_{ab}}$ and $\textcolor{teal}{\tilde{g}_{ab}}$; in case (3), the intersection occurs between the null cones of $\textcolor{navy}{g_{ab}}$ and $\textcolor{DarkRed}{\hat{g}_{ab}}$; and in case (4), between those of $\textcolor{teal}{\tilde{g}_{ab}}$ and $\textcolor{DarkRed}{\hat{g}_{ab}}$. Under certain conditions explained in the text, cases (1), (2), and (3) are strongly hyperbolic, whereas case (4) is intrinsically weakly hyperbolic. We emphasize that, since our discussion focuses on hyperbolicity, we do not impose causality restrictions. As a result, the null cones of $\textcolor{teal}{\tilde{g}_{ab}}$ and $\textcolor{DarkRed}{\hat{g}_{ab}}$ can be larger than the spacetime null cone of $\textcolor{navy}{g_{ab}}$, as shown in the figure.

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • Theorem 1
  • Remark 6
  • Remark 7
  • ...and 7 more