Invariance and stability of the profile equations of geometric optics

The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L2 the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at (y, dϕ(y)) where ϕ is the background phase. We prove that the propagation cones of the profile equations are subsets of the propagation cones of the background operator.