From the text: We consider one-dimensional inverse scattering problems (ISP) related to the equation of oscillations of an inhomogeneous medium
\[ \sigma(x)W_{tt} = (\sigma(x)W_x)_x, \quad x>0, \tag{1} \]
where \(\sigma(x)\), \(\sigma(x) > 0\), is the acoustic impedance, and assume that \(\sigma(0) = 1\). We reduce Eq. (1) to two objects to be investigated directly. The first object is the Klein-Gordon equation for a string
\[ w_{tt} = w_{xx} - q(x)w,\quad x > 0, \tag{2} \]
where the potential \(q(x) = z' + z^2\) is determined by the scattering coefficient \(z(x) = \sigma'/(2\sigma)\). Let \(z(0) = h\). It is obvious that Eq. (2) can also be considered for an arbitrary continuous potential \(q(x)\), i.e., for a potential that cannot be represented in the form indicated above.
The other object of our study is the one-dimensional nonstationary Dirac system
\[ v_t + v_x + z(x)u = 0,\quad u_t - u_x - z(x)v = 0,\quad x>0. \tag{3} \]
The Riemannian invariants \(v\) and \(u\) have the meaning of incident and scattered waves. We note that both Eqs. (2) and (3) also describe certain known quantum mechanical processes.
In this paper, we discuss the conditions of local solvability of the ISP for Eqs. (2) and (3). These problems consist in reconstructing the continuous functions \(q(x)\) and \(z(x)\) from the scattering data.