Numerical study of blow-up and stability of line solitons for the Novikov–Veselov equation

Kazeykina, Anna; Klein, Christian

doi:10.1088/1361-6544/aa6f29

Cited by 9 publications

(11 citation statements)

References 37 publications

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“…Our goal here is to study the time evolution of initial data that are not single peaked, but have the same maximum along a certain set, for example, an interval or a curve. In the wall-type data above (22) we have the maximum elongated in the y and z directions, and in the x-direction the maximum is still localized in a single point (at x = 0). The ZK evolution for such data with A = 3.6, a = 1.5 at t = 0.05, 0.2, 0.35 is shown in Fig.…”

Section: Soliton Resolutionmentioning

confidence: 75%

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Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

Klein,

Roudenko,

Stoilov

2020

Preprint

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We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This equation is L 2 -subcritical, and thus, solutions exist globally, for example, in the H 1 energy space.We first study stability of solitons with various perturbations in sizes and symmetry, and show asymptotic stability and formation of radiation, confirming the asymptotic stability result in [12] for a larger class of initial data. We then investigate the solution behavior for different localizations and rates of decay including exponential and algebraic decays, and give positive confirmation toward the soliton resolution conjecture in this equation. Finally, we investigate soliton interactions in various settings and show that there is both a quasi-elastic interaction and a strong interaction when two solitons merge into one, in all cases always emitting radiation in the conic-type region of the negative x-direction.

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Section: Soliton Resolutionmentioning

confidence: 75%

“…Figure 13. Soliton resolution and control parameters for the wall-type initial data (22). The profile of the solution at t = 5 (top left), time dependence of the L ∞ norm (top right), the difference of the solution at t = 5 and a rescaled soliton (bottom left), the Fourier coefficients at t = 5 (bottom right).…”

Section: Soliton Resolutionmentioning

confidence: 99%

Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

Klein,

Roudenko,

Stoilov

2020

Preprint

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“…18 indicate that it is resolved to the order of the rounding error. This means that (similar to the case of blow-up in the Novikov-Veselov equation [11]) the resolution is first lost in time (compare this to the case of blow-up solutions in DS II system [17], where the limiting factor is spatial resolution). The loss of resolution in time leads eventually to a breaking of the code.…”

Section: The L 2 -Critical Casementioning

confidence: 94%

Numerical study of Zakharov-Kuznetsov equations in two dimensions

Klein,

Roudenko,

Stoilov

2020

Preprint

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We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the L 2 -subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the L 2 -critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving self-similar type rescaled profile with the blow-up happening at infinity in the critical case and at a finite location in the supercritical case. In the L 2 -critical case, the blowup appears to be similar to the one in the L 2 -critical generalized Korteweg-de Vries equation with the profile being a dynamically rescaled soliton.

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“…Let φ q be a positive solution to the Schrodinger equation and let c ∞ be given by (18). By Nachman [28, lemma 3.5] for critical potentials and Music [26, theorem 1.4] for subcritical potentials with c ∞ = 0, we have that [I + G 0 * (q • )] is invertible for critical/subcritical q on W 1,p −β (R 2 ) for β > 2/p.…”

Section: Topology Of the Set Of Subcritical Potentialsmentioning

confidence: 99%

“…Moreover, the NV equation exhibits many dynamical phenomena that also occur for the KP equations. For example, the NV equation at nonzero energy has line soliton solutions and algebraic solitons (see [18] for a numerical study of the stability of such structures, their relationship with analogous structures for the KP I and KP II equations, and a review of the literature), while the NV equation at zero energy has a static lump solution. Thus, the zero-energy NV equation is part of a continuum of dispersive equations which exhibit important nonlinear wave phenomena and whose limits describe phenomena of physical interest.…”

Section: Introductionmentioning

confidence: 99%

Global Solutions for the zero-energy Novikov–Veselov equation by inverse scattering

Music

Perry

2018

Nonlinearity

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Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q 0 with the property that the associated Schrödinger operator −∂ x ∂ x + q 0 is nonnegative. Such initial data are either critical (an arbitrarily small perturbation of the potential makes the operator nonpositive) or subcritical (sufficiently small perturbations of the potential preserve non-negativity of the operator). Previously, Lassas, Mueller, Siltanen and Stahel proved global existence for critical potentials, also called potentials of conductivity type. We extend their results to include the much larger class of subcritical potentials. We show that the subcritical potentials form an open set and that the critical potentials form the nowhere dense boundary of this open set. Our analysis draws on previous work of the first author and on ideas of Grinevich and Manakov.

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Numerical study of blow-up and stability of line solitons for the Novikov–Veselov equation

Cited by 9 publications

References 37 publications

Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

Numerical study of Zakharov-Kuznetsov equations in two dimensions

Global Solutions for the zero-energy Novikov–Veselov equation by inverse scattering

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