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 Fr\'{e}chet 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 $2\times2$ matrix Schr\"odinger 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.