Roman Novikov

Explicit Formulas and Global Uniqueness for Phaseless Inverse Scattering in Multidimensions

We consider phaseless inverse scattering for the Schrödinger equation with compactly supported potential in dimension

Global uniqueness and reconstruction for the multi-channel Gelʼfand–Calderón inverse problem in two dimensions

d\ge 2

Large time asymptotics for the Grinevich-Zakharov potentials

d≥2. We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we give also a global uniqueness result for this problem with appropriate data on a fixed energy neighborhood.

New Global Stability Estimates for Monochromatic Inverse Acoustic Scattering

Abstract In this Letter we show that the Novikov–Veselov equation (NV-equation) at positive energy (an analog of KdV in 2 + 1 dimensions) has no exponentially localized solitons in the two-dimensional sense.

A LARGE TIME ASYMPTOTICS FOR TRANSPARENT POTENTIALS FOR THE NOVIKOV–VESELOV EQUATION AT POSITIVE ENERGY

Abstract We study the multi-channel Gelʼfand–Calderon inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation − Δ ψ + v ( x ) ψ = 0 , x ∈ D , where v is a smooth matrix-valued potential defined on a bounded planar domain D . We give an exact global reconstruction method for finding v from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.

Absence of exponentially localized solitons for the Novikov–Veselov equation at negative energy

Abstract We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with contour singularity in spectral parameter at a fixed real energy.

Riemann--Hilbert problem approach for two-dimensional flow inverse scattering

Abstract In this article we show that the large time asymptotics for the Grinevich–Zakharov rational solutions of the Novikov–Veselov equation at positive energy (an analog of KdV in 2 + 1 dimensions) is given by a finite sum of localized travel waves (solitons).

Moutard transform for the generalized analytic functions

We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33 (2001), pp. 670...