Stable determination of coefficients for the nonlinear third-order acoustic equations
Abstract
In this paper, we study an inverse boundary value problem for a generalized Jordan-Moore-Gibson-Thompson equation with Westervelt-type nonlinearity and space-dependent coefficients. This third-order (in time) hyperbolic equation models nonlinear ultrasound propagation in viscous and thermally relaxing media. Assuming that the diffusivity and the sound speed are known a priori, we investigate the simultaneous stable determination of the friction coefficient, the weak damping coefficient, the potential, and the nonlinear coefficient from the associated Dirichlet-to-Neumann map. The proof combines the finite-difference linearization method with the construction of Gaussian beam and geometric optics solutions. These arguments reduce the inverse problem to stability estimates for (attenuated) geodesic ray transforms under the foliation condition. As a consequence, we obtain Hölder-type stability estimates of recovering the linear and nonlinear coefficients.
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BibTeX
@misc{fu2026stable,
title = {{Stable determination of coefficients for the nonlinear third-order acoustic equations}},
author = {Song-Ren Fu and Dong Qiu and Tianyi Zheng and Ting Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12974},
eprint = {2609.12974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12974},
}
RIS
TY - UNPB
TI - Stable determination of coefficients for the nonlinear third-order acoustic equations
AU - Song-Ren Fu
AU - Dong Qiu
AU - Tianyi Zheng
AU - Ting Zhou
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12974
UR - https://arxiv.org/abs/2609.12974
AB - In this paper, we study an inverse boundary value problem for a generalized Jordan-Moore-Gibson-Thompson equation with Westervelt-type nonlinearity and space-dependent coefficients. This third-order (in time) hyperbolic equation models nonlinear ultrasound propagation in viscous and thermally relaxing media. Assuming that the diffusivity and the sound speed are known a priori, we investigate the simultaneous stable determination of the friction coefficient, the weak damping coefficient, the potential, and the nonlinear coefficient from the associated Dirichlet-to-Neumann map. The proof combines the finite-difference linearization method with the construction of Gaussian beam and geometric optics solutions. These arguments reduce the inverse problem to stability estimates for (attenuated) geodesic ray transforms under the foliation condition. As a consequence, we obtain Hölder-type stability estimates of recovering the linear and nonlinear coefficients.
ER -