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 Zh. Mat. Fiz. Anal. Geom., 2010, Volume 6, Number 1, Pages 21–33 (Mi jmag139)

A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation

I. Egorovaa, G. Teschlbc

a Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkiv, 61103, Ukraine
b International Erwin Schrödinger Institute for Mathematical Physics, 9 Boltzmanngasse, 1090, Wien, Austria
c University of Vienna

Abstract: A one-dimensional Schrödinger operator which is a short-range perturbation of a finite-gap operator is considered. There are given the necessary and sufficient conditions on the left/right reflection coefficient such that the difference of the potentials has finite support to the left/right, respectively. Moreover, these results are applied to show a unique continuation type result for solutions of the Korteweg–de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg–de Vries equation is also obtained.

Key words and phrases: inverse scattering, finite-gap background, KdV, nonlinear Paley–Wiener Theorem.

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Bibliographic databases:
MSC: Primary 34L25, 35Q53; Secondary 35B60, 37K20
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Citation: I. Egorova, G. Teschl, “A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation”, Zh. Mat. Fiz. Anal. Geom., 6:1 (2010), 21–33

Citation in format AMSBIB
\Bibitem{EgoTes10} \by I.~Egorova, G.~Teschl \paper A Paley--Wiener theorem for periodic scattering with applications to the Korteweg--de~Vries equation \jour Zh. Mat. Fiz. Anal. Geom. \yr 2010 \vol 6 \issue 1 \pages 21--33 \mathnet{http://mi.mathnet.ru/jmag139} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2655762} \zmath{https://zbmath.org/?q=an:1210.34126} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000275162600002} \elib{http://elibrary.ru/item.asp?id=13013553} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Rybkin A., “Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line”, Nonlinearity, 23:5 (2010), 1143–1167
2. Egorova I., Teschl G., “On the Cauchy Problem For the Modified Korteweg-de Vries Equation With Step Like Finite-Gap Initial Data”, Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Contemporary Mathematics, 526, eds. Holden H., Karlsen KH., Amer Mathematical Soc, 2010, 151–158
3. Rybkin A., “The Hirota tau-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line”, Nonlinearity, 24:10 (2011), 2953–2990
4. Grunert K., “The transformation operator for Schrodinger operators on almost periodic infinite-gap backgrounds”, J. Differ. Equ., 250:8 (2011), 3534–3558
5. Mikikits-Leitner A., Teschl G., “Trace Formulas For Schrodinger Operators in Connection With Scattering Theory For Finite-Gap Backgrounds”, Spectral Theory and Analysis, Operator Theory Advances and Applications, 214, eds. Janas J., Kurasov P., Laptev A., Naboko S., Stolz G., Birkhauser Verlag Ag, 2011, 107–124
6. Bennewitz C., Brown B.M., Weikard R., “A uniqueness result for one-dimensional inverse scattering”, Math. Nachr., 285:8–9 (2012), 941–948
7. Mikikits-Leitner A., Teschl G., “Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions”, J. Anal. Math., 116:907VB (2012), 163–218
8. Grunert K., “Scattering Theory For Schrodinger Operators on Steplike, Almost Periodic Infinite-Gap Backgrounds”, J. Differ. Equ., 254:6 (2013), 2556–2586
9. I. Egorova, Z. Gladka, T. L. Lange, G. Teschl, “Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials”, Zhurn. matem. fiz., anal., geom., 11:2 (2015), 123–158
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