Riemann-Hilbert problem approach for two-dimensional flow inverse scattering (Q2928114)

Inverse scattering problem for time-harmonic wave equation with first-order perturbation in two dimensions is considered. Namely, consider the equation NEWLINE\[NEWLINE -\Delta \psi -2i A(x) \nabla \psi + V(x) \psi = E \psi, \;\;x \in R^2, \;\;E>0. NEWLINE\]NEWLINE Here \(A=(A_1, A_2)\) and \(V\) are vector and scalar potentials on \(\mathbb R^2\), respectively, under some additional conditions. The above equation can be considered as a model equation for the time-harmonic acoustic pressure \(\psi\) in a two-dimensional moving fluid. The equation can also be considered as the two-dimensional Schrödinger equation at fixed energy \(E\) with the appropriate magnetic and electric potentials. The inverse problem is, given scattering amplitude at fixed \(E>0\), find potentials \(A\) and \(V\). This problem arises in particular in the acoustic tomography of moving fluid. Linearized and nonlinearized reconstruction algorithms are suggested for the problem of inverse scattering. The nonlinearized reconstruction algorithm under consideration is based on the non-local Riemann-Hilbert problem approach. Comparisons with preceding results are given.