Inverse boundary value problems of determining nonlinear coefficients for the JMGT equation

TL;DR

This paper proves that the boundary measurements uniquely determine certain nonlinear acoustic coefficients in the JMGT equation, using advanced geometric and analytical techniques.

Contribution

It establishes uniqueness results for inverse boundary value problems for the JMGT equation with nonlinearities of Kuznetsov and Westervelt types, in both Euclidean and Riemannian settings.

Findings

Boundary Dirichlet-to-Neumann map determines nonlinear coefficients uniquely.

Results hold for both Euclidean space and Riemannian manifolds under geometric assumptions.

Proof employs second order linearization and geometric optics solutions.

Abstract

We consider inverse boundary value problems for the Jordan-Moore-Gibson-Thompson (JMGT) equation in nonlinear acoustics with quadratic nonlinearities of Kuznetsov-type and Westervelt-type. We show that the associated boundary Dirichlet-to-Neumann map uniquely determines the nonlinear coefficients $\beta$ in the Westervelt-type model, and the pair $(\beta,\kappa)$ in the Kuznetsov-type model, provided that the observation time is greater than the maximal boundary-to-boundary geodesic travel time. The results are obtained in both the Euclidean setting and on compact Riemannian manifolds with proper geometric assumptions. The proof is based on the idea of second order linearization combined with the construction of geometric optics and Gaussian beam solutions, reducing the inverse problem of uniqueness to the injectivity of associated geodesic ray transforms.