We will describe generalizations of linear geometrical optics
that apply to nonlinear PDEs. The result is typically an eikonal
equation for the phase of the wave, as in linear geometrical
optics, and a nonlinear PDE for the amplitude of the wave, which
generalizes the transport equation of linear geometrical
optics. Physical applications include hyperbolic systems of
conservation laws, such as the compressible Euler equations, and
variational systems of hyperbolic PDEs, such as the Einstein
field equations in general relativity. We will also discuss the
difficulties that arise in nonlinear geometrical optics when
waves focus at a caustic.
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