Welcome to the page of Nikolay Tzvetkov







UMPA, ENS Lyon, 46, allée d'Italie, 69 007 Lyon, FRANCE

I am professor of mathematics at ENS Lyon since September 2022.
Between 2009 and 2022, I was professor at CY Cergy Paris University.
Between 2004 and 2009, I was professor at the University of Lille 1.
I was CNRS junior researcher at Orsay University in the period 2000-2004.
I defended a PhD thesis in 1999, under the supervision of Professor Jean-Claude Saut.

email: nikolay.tzvetkov (at) ens-lyon.fr

Publications

Here you can find most of my articles since 2004.

Slides of talks

NLS on product spaces and applications.
Here you can find the notes of a talk on the same topic given at Ecole Polytechnique in March 2014.

Multi-solitons for the water-waves.

On the long time behavior of the Benjamin-Ono equation.

Transverse stability issues in Hamiltonian PDE. The proceeding paper.

Solving nonlinear PDE's in the presence of low regularity randomness. This talk is based on the following survey article.

On the macroscopical description of the flow of nonlinear wave equations.

Transport of gaussian measures under Hamiltonian PDE's.

Concerning the pathological set in the context of probabilistic well-posedness.

Lectures at University of Pisa, September 2022.

Main results

My research field is partial differential equations and more precisely the mathematical analysis of nonlinear waves. I am particularly interested in regularity issues and the associated local and global well-posedness results. I also study the statistical description of such waves. I study stability of particular solutions such as solitary waves. Below, I try to give an informal description of the main achievements of my research activity :

1. Proof of the transverse instability of the water line solitary waves. This is a joint work with F. Rousset. Here you can find more details.

2. Description of the transport of gaussian measures under the flow of Hamiltonian partial differential equations. Here you can find more details.

3. Construction of an infinite sequence of invariant measures, supported by Sobolev spaces of increasing regularity, for the Benjamin-Ono equation. This is done in a series of articles, most of them joint with N. Visciglia and the final one joint with Y. Deng and N. Visciglia.

4. Probabilistic well-posedness for a nonlinear wave equation with data of super-critical Sobolev regularity. This is a joint work with N. Burq.

5. Construction and invariance of the Gibbs measure for the radial defocusing non linear Schroedinger equation, posed on a disk.

6. Construction of solutions with growing higher Sobolev norms for the defocusing non linear Schroedinger equation in three spatial dimensions. This is a joint work with Z. Hani, B. Pausader and N. Visciglia.

7. Strichartz estimates and well-posedness for the non linear Schroedinger equation, posed on a compact manifold. This is a series of joint articles with N. Burq and P. Gerard.

7+ Extension of the results in 6 to the energy critical cases in the cases of torus and spheres. This is a joint work with S. Herr and D. Tataru in the case of the torus and B. Pausader and X. Wang in the case of the sphere.

8. Failure of the semi-linear well-posedness in Sobolev spaces for the KP-I equation and the Benjamin-Ono equation. This is a joint work with L. Molinet and J.C. Saut.

9. Refined energy method for the well-posedness of the Benjamin-Ono equation. This is a joint work with H. Koch.