The Novikov-Veselov Equation: Theory and Computation

Croke, Ryan; Mueller, Jennifer L.; Music, Michael; Perry, Peter; Siltanen, Samuli; Stahel, Andreas

doi:10.48550/arxiv.1312.5427

Cited by 3 publications

(5 citation statements)

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“…We refer to the excellent survey [54] for a rather complete account of what is known in this case and which comprises some interesting numerics and a rich bibliography.…”

Section: The Novikov-veselov Equationmentioning

confidence: 99%

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IST Versus PDE: A Comparative Study

Klein

Saut

2015

Hamiltonian Partial Differential Equations and Applications

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“…We refer to the excellent survey [54] for a rather complete account of what is known in this case and which comprises some interesting numerics and a rich bibliography.…”

Section: The Novikov-veselov Equationmentioning

confidence: 99%

“…The Novikov-Veselov equation with zero energy has a global solution for critical and subcritical initial data, but its solution may blow up in finite time for supercritical initial data. 12 We refer to [54] for some partial results toward its resolution.…”

Section: The Novikov-veselov Equationmentioning

confidence: 99%

IST Versus PDE: A Comparative Study

Klein

Saut

2015

Hamiltonian Partial Differential Equations and Applications

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“…In that sense, NV is the closest model that generalizes (in a mathematical form) the famous Korteweg-de Vries (KdV) equation to the two dimensional case, and it has interesting connections with the scattering problem for bounded or localized potentials, and the theory of inverse problems for sufficiently smooth potentials, see e.g. [8] and references therein for more details. Moreover, if v depends only on x, we have z = z = x, ∂ z = ∂ z = 1 2 ∂ x and…”

Section: Introduction Main Resultsmentioning

confidence: 99%

“…In order to estimate I 1 we use the following estimates valid on ∂B ε (λ (1) ||λ − λ (2) ||λ − λ (3) ||λ − λ (4) ||λ − λ (5) ||λ − λ (6) | εω 2 . 8 The worst case corresponds to ω being close to 1, and |λ2| is chosen close to 1 This allows us to obtain…”

Section: Re λmentioning

confidence: 99%

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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“…It is a nonlinear evolution equation generalizing the celebrated Korteweg-de Vries (KdV) equation into dimension (2+1). There has been significant recent progress in linearizing the NV equation using inverse scattering methods, see [38,37,44,42,14].…”

Section: Introductionmentioning

confidence: 99%