Some Results on Inverse Scattering

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: 1) Property $C$ and applications, 2) Stable inversion of fixed-energy 3D scattering data and its error estimate, 3) Inverse scattering with ''incomplete`` data, 4) Inverse scattering for inhomogeneous Schr\"odinger equation, 5) Krein's inverse scattering method, 6) Invertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion methods, 8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, 10) Inverse obstacle scattering with fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a fixed incident direction, 12) Creating materials with a desired refraction coefficient and wave-focusing properties.Comment: 24p

A novel nonlinear evolution equation integrable by the inverse scattering method

A Backlund transformation for an evolution equation (ut+u ux)x+u=0 transformed into new coordinates is derived. An inverse scattering problem is formulated. The inverse scattering method has a third order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described

Phaseless inverse scattering in the one-dimensional case

We consider the one-dimensional Schr\"odinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for $x<0$. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension $d=1$