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Nonlinear electrodynamics as a symmetric hyperbolic system
Nonlinear theories generalizing Maxwell's electromagnetism and arising from a
Lagrangian formalism have dispersion relations in which propagation planes
factor into null planes corresponding to two effective metrics which depend on
the point-wise values of the electromagnetic field. These effective Lorentzian
metrics share the null (generically two) directions of the electromagnetic
field. We show that, the theory is symmetric hyperbolic if and only if the
cones these metrics give rise to have a non-empty intersection. Namely that
there exist families of symmetrizers in the sense of Geroch which are positive
definite for all covectors in the interior of the cones intersection. Thus, for
these theories, the initial value problem is well-posed. We illustrate the
power of this approach with several nonlinear models of physical interest such
as Born-Infeld, Gauss-Bonnet and Euler-Heisenberg
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