Uniqueness Result For Semi-linear Wave Equations With Sources
Dong Qiu, Xiang Xu, Yeqiong Ye, Ting Zhou
TL;DR
The paper addresses the inverse boundary value problem for semilinear wave equations with an internal source, aiming to recover nonlinear coefficients, the source, and initial data from the Dirichlet-to-Neumann map. It combines higher-order linearization with CGO constructions to reveal that, in general, recovery is governed by gauge-type ambiguities: for a polynomial nonlinearity of degree $n$, only the highest-order coefficient is uniquely determined, while lower-order coefficients and the source are recoverable up to a gauge involving a function $\psi$. The authors establish local well-posedness of the forward problem and prove a series of uniqueness results: in the quadratic case explicit gauge relations appear, and in the general case an inductive higher-order linearization argument shows that all derivatives of the nonlinear term match between two configurations, yielding either unique recovery or gauge equivalence depending on the nonlinearity. The results are extended to several non-polynomial nonlinearities, with a set of cases illustrating when unique identifiability holds versus when gauge symmetry persists. These findings highlight the critical influence of nonlinearity structure on identifiability in nonlinear hyperbolic inverse problems and inform potential strategies for simultaneous recovery in applications like nonlinear wave imaging.
Abstract
This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general semilinear term and an internal source. The inverse problem is to determine the nonlinear coefficients (potentials), the source term, and the initial data from the Dirichlet-to-Neumann (DtN) map. Our approach combines higher-order linearization and the construction of complex geometrical optics (CGO) solutions. The main results establish that while unique recovery is not always possible, we can precisely characterize the gauge equivalence classes in the solutions to this inverse problem. For a wave equation with a polynomial nonlinearity of degree $n$, we prove that only the highest-order coefficient can be uniquely determined from the DtN map; the lower-order coefficients and the source can only be recovered up to a specific gauge transformation involving a function $ψ$. Furthermore, we provide sufficient conditions under which unique determination of all parameters is guaranteed. We also extend these results to various specific non-polynomial nonlinearities, demonstrating that the nature of the nonlinearity critically influences whether unique recovery or a gauge symmetry is obtained.