The non-uniqueness of solution to the inverse problem of scattering (Q2760739)

The author discusses the classical model of the wave propagation process NEWLINE\[NEWLINE \begin{aligned} &\Bigl(\Delta - \frac{1}{c^2(x)}\frac{\partial^2}{\partial t^2}\Bigr)u(x,t)= \delta(x-x_0)\delta(t),\;\;t\in {\mathbb R}^1,\;x\in {\mathbb R}^3, \\ &u(x,t)=0,\;\;t<0,\;x\in {\mathbb R}^3. \end{aligned} NEWLINE\]NEWLINE It is assumed that the velocity \(c(x)\) is represented as: \(c(x)=c_0+c_1(x)\), with \(c_0\) a constant and \(c_1\) a function supported in a bounded domain \(D\) with boundary \(S\). The inverse problem consists in the reconstruction of the velocity \(c(x)\) with the use of the values of \(u(x,t)\) on some surface \(S_0\) such that the domain \(D\) is included into the interior part of \(S_0\). By applying the Fourier transform in \(t\), the problem is reduced to a system of integral equations, and it is pointed out that this system is not uniquely solvable. The article does not contain exact statements.