767 research outputs found
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
--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
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
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
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