Inverse source scattering problem for a nonlinear Schr\"odinger equation

Lei Zhang; Yue Zhao

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arXiv · 2303.11628

Inverse source scattering problem for a nonlinear Schr\"odinger equation

Lei Zhang·Yue Zhao

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Abstract

We study an inverse source scattering problem for the Schr\"odinger equation with a quadratic nonlinearity. In general, uniqueness of inverse source problems can not be guaranteed at a fixed energy. Therefore, additional information is required for the source in order to obtain a unique solution. By adding reference point sources, we show that a general source function could be uniquely determined from boundary measurements at a fixed wavenumber. This method does not apply to inverse source problems of linear equations since it uses the nonlinearity as a tool. The proof utilizes the method of linearization to reduce the nonlinear inverse source scattering problem to the inverse potential scattering problem of the linear Schr\"odinger equation.

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Lei Zhang, Yue Zhao. 2023-03-21. Inverse source scattering problem for a nonlinear Schr\"odinger equation. https://arxiv.org/abs/2303.11628

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