Decay estimates of wave equations in un-isotropic media

We prove decay estimates for solutions to non-isotropic linear systems of wave equations. The defining feature of these estimates is that they depend only on the commutation properties of the system with the scaling vector field. As application we give two surprisingly simple proofs for small data global regularity results non-isotropic systems of wave equations in ${\mathbb{R}}^{1+3}$ with cubic semilinear nonlinearities. We hope that the techniques presented here are relevant for the more difficult and important case of biaxial refraction in crystal optics.