Recovery of a general nonlinearity in the semilinear wave equation

Antônio Sá Barreto; Plamen Stefanov

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arXiv · 2107.08513

Recovery of a general nonlinearity in the semilinear wave equation

Antônio Sá Barreto·Plamen Stefanov

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Abstract

We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Δu+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $\sim h$ and amplitude $\sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $\text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $α(x)$ when $f(x,u)=α(x)u^{2m}$. This is done in an explicit way as $h\to0$.

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BibTeX RIS

Antônio Sá Barreto, Plamen Stefanov. 2021-07-18. Recovery of a general nonlinearity in the semilinear wave equation. https://arxiv.org/abs/2107.08513

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