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

TL;DR

This paper investigates an inverse problem for the semilinear wave equation, demonstrating how to recover a non-linear coefficient using harmonic wave probing and phase shift measurements, with implications for uniqueness and reconstruction.

Contribution

It introduces a method to recover the non-linear coefficient's Radon transform from phase shifts of harmonic waves in the weakly non-linear regime.

Findings

Radon transform of the non-linearity can be extracted from phase shifts.

Unique recovery of the non-linearity in the linearized case.

Method applicable in two and three dimensions.

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.