Recovery of a general nonlinearity in the semilinear wave equation

Abstract

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