Unique Determination of Sound Speeds for Coupled Systems of Semi-linear Wave Equations

TL;DR

This paper demonstrates that for coupled semi-linear wave equations with different sound speeds, the internal sound speeds can be uniquely reconstructed from boundary measurements, under certain conditions.

Contribution

It establishes the unique determination of sound speeds in coupled nonlinear wave systems from boundary data, extending previous results to more complex systems.

Findings

Reconstruction of sound speeds is possible under small initial data.

Unique determination holds under specific geometric conditions.

Results apply to both partial and full boundary measurements.

Abstract

We consider coupled systems of semi-linear wave equations with different sound speeds on a finite time interval $[0,T]$ and a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^3$, with boundary $\partial\Omega$. We show the coupled systems are well posed for variable coefficient sounds speeds and short times. Under the assumption of small initial data, we prove the source to solutions map on $[0,T]\times\partial\Omega$ associated with the nonlinear problem is sufficient to determine the source-to-solution map for the linear problem. We can then reconstruct the sound speeds in $\Omega$ for the coupled nonlinear wave equations under certain geometric assumptions. In the case of the full source to solution map in $\Omega\times[0,T]$ this reconstruction could also be accomplished under fewer geometric assumptions.