Inverse problems for a quasilinear hyperbolic equation with multiple unknowns

TL;DR

This paper establishes unique reconstruction methods for unknown coefficients and initial data in a quasilinear hyperbolic PDE on a Riemannian manifold using boundary measurements, with applications to nonlinear wave phenomena.

Contribution

It introduces a novel approach combining Gaussian beam solutions and light-ray transforms to solve inverse boundary problems for quasilinear hyperbolic equations.

Findings

Unique determination of coefficients and initial data from hyperbolic DtN map

Construction of Gaussian beam solutions for quasilinear hyperbolic PDEs

Application of light-ray transforms and stationary phase techniques

Abstract

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ 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.