Logarithmic Stability for an Inverse Source Problem in a Coupled Nonlinear Helmholtz System
Abstract
We study an inverse source problem for a two-mode nonlinear Helmholtz system motivated by second-harmonic generation. The datum is the full first Fréchet derivative of the nonlinear Dirichlet-to-Neumann map at one fixed, common, small boundary state, which need not be zero. For real-valued small data, a known susceptibility bounded away from zero, and sources supported in a fixed compact subset of the domain, we prove uniqueness and a conditional single-logarithmic stability estimate. The first boundary variation is the Dirichlet-to-Neumann map of a symmetric 2×2 matrix Schrödinger operator. Absorbing the two known Helmholtz energies into its matrix potential permits the use of standard zero-energy complex geometrical optics solutions with a common null phase geometry. A bilinear Alessandrini identity then gives a uniform Fourier estimate for the two entries containing the background fields. Zero extension below the exponent 3/2, a low/high frequency splitting, and interpolation between H−2 and the a priori Hs source bound yield the stated stability modulus.
Related
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BibTeX
@misc{ye2026logarithmic,
title = {{Logarithmic Stability for an Inverse Source Problem in a Coupled Nonlinear Helmholtz System}},
author = {Hongcan Ye and Lei Zhang and Ting Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09845},
eprint = {2609.09845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09845},
}
RIS
TY - UNPB
TI - Logarithmic Stability for an Inverse Source Problem in a Coupled Nonlinear Helmholtz System
AU - Hongcan Ye
AU - Lei Zhang
AU - Ting Zhou
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09845
UR - https://arxiv.org/abs/2609.09845
AB - We study an inverse source problem for a two-mode nonlinear Helmholtz system motivated by second-harmonic generation. The datum is the full first Fréchet derivative of the nonlinear Dirichlet-to-Neumann map at one fixed, common, small boundary state, which need not be zero. For real-valued small data, a known susceptibility bounded away from zero, and sources supported in a fixed compact subset of the domain, we prove uniqueness and a conditional single-logarithmic stability estimate. The first boundary variation is the Dirichlet-to-Neumann map of a symmetric 2×2 matrix Schrödinger operator. Absorbing the two known Helmholtz energies into its matrix potential permits the use of standard zero-energy complex geometrical optics solutions with a common null phase geometry. A bilinear Alessandrini identity then gives a uniform Fourier estimate for the two entries containing the background fields. Zero extension below the exponent 3/2, a low/high frequency splitting, and interpolation between H^−2 and the a priori H^s source bound yield the stated stability modulus.
ER -