Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

Abstract

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form $c{\left(x\right)}^{-2}{\partial }_t^{2}u={\Delta }_g(u+F(x,u\left)\right)+G(x,u)$ on a compact Riemannian manifold $(M,g)$ with boundary. We show that if $F(x,u)$ is monomial and $G(x,u)$ is analytic in u, then $F,G$ and c as well as the associated initial data can be uniquely determined and reconstructed by the corresponding hyperbolic DtN (Dirichlet-to-Neumann) map. Our work leverages the construction of proper Gaussian beam solutions for quasilinear hyperbolic PDEs as well as their intriguing applications in conjunction with light-ray transforms and stationary phase techniques for related inverse problems. The results obtained are also of practical importance in assorted applications with nonlinear waves.