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arXiv · 2306.15935

Partial Data Inverse Problems for the Nonlinear Schr\"odinger Equation

Ru-Yu Lai·Xuezhu Lu·Ting Zhou
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Abstract

In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient $\beta(t, x)$ in the Schr\"odinger equation $(i\partial_t + \Delta + q(t, x))u + \beta u^2 = 0$, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN-map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain type of geometric optics (GO) solutions can reach; and a stability estimate based on the unique continuation property for the linear equation.

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Ru-Yu Lai, Xuezhu Lu, Ting Zhou. 2023-06-28. Partial Data Inverse Problems for the Nonlinear Schr\"odinger Equation. https://arxiv.org/abs/2306.15935

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