Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime

Antônio Sá Barreto; Plamen Stefanov

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arXiv · 2102.06323

Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime

Antônio Sá Barreto·Plamen Stefanov

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Abstract

We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-\Delta u+ \alpha(x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength $h$ and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits the support of $\alpha$. We show that one can extract the Radon transform of $\alpha$ from the phase shift of such waves, and then one can recover $\alpha$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.

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Antônio Sá Barreto, Plamen Stefanov. 2021-02-12. Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime. https://arxiv.org/abs/2102.06323

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