Homogenization of systems of wave equations and ring solutions with dispersive profiles

Abstract

We consider systems of wave equations such as the timedependent Lamé system or elasticity. When the coefficients are periodic in space, the classical task in homogenization theory is to describe limits of solutions when the periodicity tends to zero. The effective equation is a system with constant coefficients, typically of the same structure as the original system. Instead, when long time intervals are considered, new dispersive terms can appear in the effective system. We derive such dispersive effective systems of wave equations using the Bloch method of homogenization. The method yields approximate representation formulas for solutions in Fourier space. These also allow to describe solutions as superpositions of ring waves, expanding with constant speed, with profiles that change on a slow time scale according to the dispersive terms.