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 $c(x{)}^{-2}{\partial }_t^{2}u=\hat{I}{”}_g(u+F(x,u))+G(x,u)$ 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 of applications with nonlinear waves.