77 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
$d$--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

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

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

### Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

We prove the existence of small amplitude periodic solutions, for a large
Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak
quadratic and velocity dependent nonlinearity and with Dirichlet boundary
conditions. Such nonlinear PDE can be regarded as a simple model describing
oscillations of flexible structures like suspension bridges in presence of an
uniform wind flow. The periodic solutions are explicitly constructed by means
of a perturbative expansion which can be considered the analogue of the
Lindstedt series expansion for the invariant tori in classical mechanics. The
periodic solutions are not analytic but defined only in a Cantor set, and
resummation techniques of divergent powers series are used in order to control
the small divisors problem.Comment: 29 pages 6 figure

### KAM theory in configuration space and cancellations in the Lindstedt series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian
systems yields that the perturbation expansion (Lindstedt series) for
quasi-periodic solutions with Diophantine frequency vector converges. If one
studies the Lindstedt series, one finds that convergence is ultimately related
to the presence of cancellations between contributions of the same perturbation
order. In turn, this is due to symmetries in the problem. Such symmetries are
easily visualised in action-angle coordinates, where KAM theorem is usually
formulated, by exploiting the analogy between Lindstedt series and perturbation
expansions in quantum field theory and, in particular, the possibility of
expressing the solutions in terms of tree graphs, which are the analogue of
Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying
terms theorem ensures that an analytic quasi-periodic solution with the same
Diophantine frequency vector as the unperturbed Hamiltonian exists for the
system obtained by adding a suitable constant (counterterm) to the vector
field. Also in this case, one can follow the alternative approach of studying
the perturbation expansion for both the solution and the counterterm, and again
convergence of the two series is obtained as a consequence of deep
cancellations between contributions of the same order. We revisit Moser's
theorem, by studying the perturbation expansion one obtains by working in
Cartesian coordinates. We investigate the symmetries giving rise to the
cancellations which makes possible the convergence of the series. We find that
the cancellation mechanism works in a completely different way in Cartesian
coordinates. The interpretation of the underlying symmetries in terms of tree
graphs is much more subtle than in the case of action-angle coordinates.Comment: 38 pages, 18 fugure

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