Large time asymptotics for the Grinevich–Zakharov potentials

Kazeykina, Anna; Novikov, Roman

doi:10.1016/j.bulsci.2011.02.003

Cited by 14 publications

(16 citation statements)

References 7 publications

(3 reference statements)

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“…It was shown in [N2] that all soliton-type (traveling wave) solutions of (1.1) with E > 0 must have a zero scattering amplitude at fixed energy; in addition it was proved in [N2] that for the equation (1.1) with E > 0 no exponentially localized soliton-type solutions exist (even if the scattering data are allowed to have singularities). However, in [G1], [G2] a family of algebraically localized solitons (traveling waves) was constructed de facto (see also [KN2]). We note that for the case E < 0, though the absence of exponentially-localized solitons has been proved (see [KN3]), the existence of bounded algebraically localized solitons is still an open question.…”

Section: Introductionmentioning

confidence: 99%

A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data

Kazeykina¹

2012

Inverse Problems

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Abstract. In the present paper we are concerned with the Novikov-Veselov equation at negative energy, i.e. with the (2 + 1)-dimensional analog of the KdV equation integrable by the method of inverse scattering for the two-dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non-singular scattering data behaves asymptotically as const t 3/4 in the uniform norm at large times t. We also prove that this asymptotics is optimal.

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Section: Introductionmentioning

confidence: 99%

A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data

Kazeykina¹

2012

Inverse Problems

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“…We also refer to [99] and to the papers [100,101,102,103,104,105,177] for many interesting results on the soliton solutions (absence, decay properties,..) and the long time asymptotics. More precisely it is proven that the rational, non singular solutions introduced in [79,80] at positive energy are multi-solitons.…”

Section: 22mentioning

confidence: 99%

IST Versus PDE: A Comparative Study

Klein

Saut

2015

Hamiltonian Partial Differential Equations and Applications

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“…However, the existence of global solutions for large initial data is still an open problem. At positive energy, N V + exhibits a regime similar in some aspects to that of KPI (see [14,15,17] for results on N V + and [28,27,5,6] for related results on KPI). Note that from the point of view of behavior of the solution to the Cauchy problem, KPI is essentially different from KPII: KPI is essentially a quasilinear equation, while KPII is semilinear.…”

Section: The Nv Equationmentioning

confidence: 93%

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

Kazeykina

Muñoz

2016

Journal of Functional Analysis

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We continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method and X s,b spaces to obtain local well-posedness of NV at positive energy in H s , s > 1 2 . Our result implies, in particular, that at least for s > 1 2 , NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations.As a supplement, we provide some new explicit solutions of NV at zero energy which exhibit an interesting behaviour at large times.

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Large time asymptotics for the Grinevich–Zakharov potentials

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).

Cited by 14 publications

References 7 publications

A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data

A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data

IST Versus PDE: A Comparative Study

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

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