Recovering initial values from light cone traces of solutions of the wave equation

Abstract

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R n .

Reconstruction of an initial function from the solutions of the fractional wave equation on the light cone trace

“…That is,  𝛼 assigns the traces of the solution p on the light cones t = |x| to the initial data 𝑓 . Rakesh and Yuan [4] derived the inversion of this linear operator 𝑓  →  𝛼 𝑓 only for an odd dimension n with 𝛼 = 2. They identified the relationship between  𝛼 𝑓 and the Radon transform and used it to obtain the inverse transform formula.…”

“…Herein, we aim to reconstruct 𝑓 from  𝛼 𝑓 for any dimension n with 1 < 𝛼 ≤ 2. Rakesh and Yuan [4] report that the problem of inverting  2 𝑓 is related to the exterior Goursat problem and the general relativity (see [5][6][7]). An event occurring in spacetime causes the emission of light.…”

“…$$ That is, $$$ {\mathcal{W}}_{\alpha } $$$ assigns the traces of the solution $$$ p $$$ on the light cones $$$ t&amp;amp;#x0003D;\mid \mathbf{x}\mid $$$ to the initial data $$$ f $$$. Rakesh and Yuan [4] derived the inversion of this linear operator $$$ f\mapsto {\mathcal{W}}_{\alpha }f $$$ only for an odd dimension $$$ n $$$ with $$$ \alpha &amp;amp;#x0003D;2 $$$. They identified the relationship between $$$ {\mathcal{W}}_{\alpha }f $$$ and the Radon transform and used it to obtain the inverse transform formula.…”

“…Rakesh and Yuan [4] report that the problem of inverting $$$ {\mathcal{W}}_2f $$$ is related to the exterior Goursat problem and the general relativity (see [5–7]). An event occurring in spacetime causes the emission of light.…”

Uniqueness for time-dependent inverse problems with single dynamical data

“…We consider in this work the uniqueness result for simultaneously recovering both the sound speed and its convex polygonal support from a single dynamical data. For more related works, we refer to [1,18,19,34,35,36,37,38,39,46].…”