Reducible quasi-periodic solutions for the Non Linear Schr\"odinger equation

The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a $d$--dimensional torus \T^d. The main point is to prove that prove that the normal form is reducible, block diagonal and satisfies the second Melnikov condition block wise. From this we deduce the result by a KAM algorithm.Comment: 48 page

On the theorem of Amitsur--Levitzki

We present a proof of the Amitsur--Levitzki theorem which is a basis for a general theory of equivariant skew--symmetric maps on matrices.Comment: 3page

Growth of Sobolev norms for the quintic NLS on T2\mathbb T^2

We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:0808.1742 by other author

Conservation of resonant periodic solutions for the one-dimensional nonlinear Schroedinger equation

We consider the one-dimensional nonlinear Schr\"odinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the zero-mass term). We investigate conservation of small amplitude periodic-solutions for a large set measure of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough and close enough to each other (wave packets with large wave number)

Periodic solutions for the Schroedinger equation with nonlocal smoothing nonlinearities in higher dimension

We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.Comment: 60 pages 8 figure