Inverse problems for coupled wave equations with multiple sound speeds

We begin with a review of well-posedness results for the inhomogeneous wave equation with variable coefficients on a bounded domain in ${\mathbb{R}}^3$. After the warm up, we consider coupled systems of semi-linear wave equations and discuss the notion of short time well-posedness for the system. Under the assumption of small Cauchy data, we show the source to solution map for the nonlinear problem determines the source to solution map for the linear problem. We discuss why the assumption of small data and short times allows us to recover multiple sound speeds uniquely for the nonlinear problem in some natural settings. A variety of open problems are mentioned during the talk.