AI 中文总结
本文针对二次谐波产生的双模非线性亥姆霍兹系统，利用Dirichlet-to-Neumann映射的一阶导数，证明了反源问题的唯一性和单对数稳定性估计。
AI 中文摘要
我们研究了一个由二次谐波产生所启发的双模非线性亥姆霍兹系统的反源问题。数据是非线性Dirichlet-to-Neumann映射在某个固定的、共同的、小的边界状态（该状态不必为零）处的全一阶Fréchet导数。对于实值小数据、一个已知的远离零的磁化率，以及支撑在域内固定紧子集中的源，我们证明了唯一性和一个条件性的单对数稳定性估计。第一个边界变分是对称的$2\ imes2$矩阵薛定谔算子的Dirichlet-to-Neumann映射。将两个已知的亥姆霍兹能量吸收到其矩阵势中，允许使用具有共同零相位几何的标准零能量复几何光学解。一个双线性Alessandrini恒等式随后给出了包含背景场的两个分量的均匀傅里叶估计。指数$3/2$以下的零延拓、低/高频分裂以及$H^{-2}$与先验$H^s$源界之间的插值产生了所陈述的稳定性模量。
英文摘要
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 $2\times2$ 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.
