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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 3(447), Pages 3–16 (Mi umn9867)  

Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials

A. D. Agaltsova, R. G. Novikovbc

a Max-Planck-Institut für Sonnensystemforschung, Göttingen, Germany
b Institute of Earthquake Prediction Theory and Mathematical Geophysics of Russian Academy of Sciences
c École Polytechnique, Centre de Mathématiques Appliquées, Palaiseau, France

Abstract: The inverse scattering problem is considered for the two-dimensional Schrödinger equation at fixed positive energy. The results include inverse scattering reconstructions from the simplest scattering amplitudes. In particular, a complete analytic solution is given of the phased and phaseless inverse scattering problems for single-point potentials of Bethe–Peierls–Fermi–Zeldovich–Berezin–Faddeev type. Numerical inverse scattering reconstructions from the simplest scattering amplitudes are then studied using the method of the Riemann–Hilbert–Manakov problem in soliton theory. Finally, these numerical inverse scattering results are used to construct corresponding numerical solutions of the non-linear equations of the Novikov–Veselov hierarchy at fixed positive energy.
Bibliography: 21 titles.

Keywords: inverse scattering, Schrödinger equation, numerical analysis, Novikov–Veselov equation.

DOI: https://doi.org/10.4213/rm9867

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English version:
Russian Mathematical Surveys, 2019, 74:3, 373–386

Bibliographic databases:

UDC: 517.958
MSC: 35J10, 35P25, 35R30
Received: 19.12.2018

Citation: A. D. Agaltsov, R. G. Novikov, “Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials”, Uspekhi Mat. Nauk, 74:3(447) (2019), 3–16; Russian Math. Surveys, 74:3 (2019), 373–386

Citation in format AMSBIB
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\paper Examples of solution of the inverse scattering problem and the equations of the Novikov--Veselov hierarchy from the scattering data of point potentials
\jour Uspekhi Mat. Nauk
\yr 2019
\vol 74
\issue 3(447)
\pages 3--16
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\jour Russian Math. Surveys
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\pages 373--386
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