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Nikolay Tzvetkov

Publications and source records attributed to Nikolay Tzvetkov.

At least 73 records · Page 4Linked to original sources
arXiv2013

On the propagation of weakly nonlinear random dispersive waves

We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast Physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data.

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arXiv2013

Invariant measures and long time behaviour for the Benjamin-Ono equation II

As a continuation of our previous work on the subject, we prove new measure invariance results for the Benjamin-Ono equation. The measures are associated with conservation laws whose leading term is a fractional Sobolev norm of order larger or equal than 5/2. The new ingredient, compared with the case of conservation laws whose leading term is an integer Sobolev norm of order larger or equal than 3 (that has been studied in our previous work), is the use of suitable orthogonality relations satisfied by multilinear products of centered complex independent gaussian variables. We also give some partial results for the measures associated with the two remaining conservation laws at lower regularity. We plan to complete the proof of their invariance in a separated article which will be the final in the series. Finally in an appendix, we give a brief comparing of the recurrence properties of the flows of Benjamin-Ono and KdV equations.

math.AP↗
arXiv2013

Invariant measures and long time behaviour for the Benjamin-Ono equation

We study the Benjamin-Ono equation, posed on the torus. We prove that an infinite sequence of weighted gaussian measures, constructed in our previous work, are invariant by the flow of the equation. These measures are supported by Sobolev spaces of increasing regularities. As a by product we deduce informations on the long time behaviour of regular solutions. To our knowledge this is the first result which gives an evidence about recurrence properties of the Benjamin-Ono equation flow.

math.AP↗
arXiv2013

Global regularity for the energy-critical NLS on $\mathbb{S}^3$

We establish global existence for the energy-critical nonlinear Schrödinger equation on $\mathbb{S}^3$. This follows similar lines to the work on $\mathbb{T}^3$ but requires new extinction results for linear solutions and bounds on the first nonlinear iterate at a Euclidean profile that are adapted to the new geometry.

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arXiv2012

The NLS ground states on product spaces

We study the nature of the Nonlinear Schrödinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.

math.AP↗
arXiv2011

Stability and instability of the KdV solitary wave under the KP-I flow

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2π\mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0 c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.

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arXiv2011

Probabilistic well-posedness for the cubic wave equation

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs measures), and second, we show that the flow that we are able to construct enjoys very nice dynamical properties, including a new notion of probabilistic continuity.

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arXiv2010

Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H^1(T^3)

A refined trilinear Strichartz estimate for solutions to the Schrödinger equation on the flat rational torus T^3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schrödinger Equation in H^s(T^3) for all s \geq 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

math.AP↗
arXiv2010

Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications

We consider the energy critical nonlinear Schrödinger equation on periodic domains of the form R^m x T^{4-m} with m=0,1,2,3. Assuming that a certain L^4 Strichartz estimate holds for solutions to the corresponding linear Schrödinger equation, we prove that the nonlinear problem is locally well-posed in the energy space. Then we verify that the L^4 estimate holds if m=2,3, leaving open the cases m=0,1.

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arXiv2010

Global well-posedness for the KP-II equation on the background of a non localized solution

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations : perturbations that are square integrable in $ \R\times \T $ and perturbations that are square integrable in $ \R^2 $. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.

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arXiv2010

Long time dynamics for the one dimensional non linear Schrödinger equation

In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).

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