  This paper argues that curvelets provide a powerful tool for representing
very general linear symmetric systems of hyperbolic differential equations.
Curvelets are a recently developed multiscale system in which the elements are
highly anisotropic at fine scales, with effective support shaped according to
the parabolic scaling principle width ~ length^2 at fine scales. We prove that
for a wide class of linear hyperbolic differential equations, the curvelet
representation of the solution operator is both optimally sparse and well
organized. It is sparse in the sense that the matrix entries decay nearly
exponentially fast (i.e. faster than any negative polynomial), and
well-organized in the sense that the very few nonnegligible entries occur near
a shifted diagonal. Indeed, we actually show that the action of the wave-group
on a curvelet is well-approximated by simply translating the center of the
curvelet along the Hamiltonian flow--hence the diagonal shift in the curvelet
epresentation. A physical interpretation of this result is that curvelets may
be viewed as coherent waveforms with enough frequency localization so that they
behave like waves but at the same time, with enough spatial localization so
that they simultaneously behave like particles.
