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Solitons and large time asymptotics of solutions for the Novikov-Veselov equation

Abstract : This work is concerned with the study of the Novikov-Veselov equation, a ( 2 + 1 )-dimensional analog of the renowned Korteweg-de Vries equation, integrable via the inverse scattering transform for the 2-dimensional stationary Schrödinger equation at a fixed energy. We start by studying a special class of rational nonsingular algebraically localized solutions of the Novikov-Veselov equation at positive energy constructed by Grinevich and Zakharov and we demonstrate that these solutions are multisolitons. Grinevich-Zakharov solutions are localized as $ O( | x |^{ -2 } ) $, $ | x | \to \infty $, and in the present work we prove that this localization is almost the strongest possible: we show that the Novikov-Veselov equation at nonzero energy does not possess solitons localized stronger than $ O( | x |^{ - 3 } ) $, $ | x | \to \infty $. For the case of zero energy we show that if the solitons of the Novikov-Veselov equation belong to the range of solutions of the modified Novikov-Veselov equation under Miura transform, then localization stronger than $ O( | x |^{ -2 } ) $ is not possible. In the present work we also study the question of the asymptotic behavior of solutions to the Cauchy problem for the Novikov-Veselov equation at nonzero energy (for the case of positive energy transparent or reflectionless solutions are considered). Under assumption that the scattering data for the solutions are nonsingular we obtain that these solutions decrease uniformly with time as $ O( t^{ -1 } ) $, $ t \to +\infty $, in the case of positive energy and as $ O( t^{ -3/4 } ) $, $ t \to +\infty $, in the case of negative energy; in the latter case we also demonstrate that the obtained estimate is optimal.
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Contributor : Anna Kazeykina <>
Submitted on : Friday, December 7, 2012 - 3:48:47 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM
Long-term archiving on: : Monday, March 11, 2013 - 11:45:40 AM

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  • HAL Id : pastel-00762662, version 1

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Anna Kazeykina. Solitons and large time asymptotics of solutions for the Novikov-Veselov equation. Analysis of PDEs [math.AP]. Ecole Polytechnique X, 2012. English. ⟨pastel-00762662⟩

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