Recovery of Zeroth Order Coefficients in Non-Linear Wave Equations

Abstract

This paper is concerned with the resolution of an inverse problem related to the recovery of a function $V$ from the source to solution map of the semi-linear equation $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}},g)$ . We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

Inverse Problem for the Yang–Mills Equations

“…The only known results are in the case G = U (1), see [12,41], and these results impose convexity assumptions not satisfied by the geometric setting of Figure 1. The same is true for recovery zeroth order terms, solved with and without convexity assumptions for certain scalar linear [43] and non-linear wave equations [14], respectively.…”

“…Therefore V (k) and V (kl) are smooth near z. Moreover, as WF(V (k) ) and WF(V (kl) ) are disjoint from the characteristic set {τ = 0} of ∂ t , the ordinary differential equation in (14) implies that also U (k) and U (kl) are smooth near z.…”

“…Recall that the temporal gauge version T (V ) of a connection V is defined by (14). Recall, furthermore, that the system (25) of Yang-Mills equations in relative Lorenz gauge with the compatibility condition is posed on M = (−2, 2) × R 3 .…”

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

“…In what follows, we refer to [20] for omitted details. To solve the eikonal equation ( 29) approximatively, one sets…”

“…. , e n ) of the tangent space T z 0 N over the single point z 0 with e 0 = V 0 was shown in [20]. By using local coordinates (x k ) on a neighbourhood U ⊂ M of z 0 let us define the mapping…”