Uniqueness of some space dependent coefficients in a wave equation of nonlinear acoustics

TL;DR

This paper proves the uniqueness of identifying space-dependent coefficients in a nonlinear acoustics wave equation using boundary pressure data, relevant for ultrasound tomography, by applying the Inverse Function Theorem.

Contribution

It introduces a novel application of the Inverse Function Theorem to establish uniqueness in parameter identification for the JMGT equation.

Findings

Proved uniqueness of space-dependent coefficients in the JMGT equation.

Established the differentiability and isomorphism of the forward operator.

Applicable to ultrasound tomography methods.

Abstract

In this paper we prove uniqueness for some parameter identification problems for the JMGT equation, a third order in time quasilinear PDE in nonlinear acoustics. The coefficients to be recovered are the space dependent nonlinearity parameter, sound speed, and attenuation parameter, and the observation available is a single time trace of the acoustic pressure on the boundary. This is a setting relevant to several ultrasound based tomography methods. Our approach relies on the Inverse Function Theorem, which requires to prove that the forward operator is a differentiable isomorphism in appropriately chosen topologies and with an appropriate choice of the excitation.