1,582 research outputs found
The Szeg\"o Cubic Equation
We consider the following Hamiltonian equation on the Hardy space on
the circle, where is the Szeg\"o
projector. This equation can be seen as a toy model for totally non dispersive
evolution equations. We display a Lax pair structure for this equation. We
prove that it admits an infinite sequence of conservation laws in involution,
and that it can be approximated by a sequence of finite dimensional completely
integrable Hamiltonian systems. We establish several instability phenomena
illustrating the degeneracy of this completely integrable structure. We also
classify the traveling waves for this system
The cubic szego equation and hankel operators
This monograph is an expanded version of the preprint arXiv:1402.1716 or
hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic
Szeg{\"o} equation on the circle ,Here denotes the orthogonal projector from onto the subspace of functions with
nonnegative Fourier modes.We construct a nonlinear Fourier transformation on
allowing to describe
explicitly the solutions of this equationwith data in . This explicit description implies
almost-periodicity of every solution in . Furthermore, it
allows to display the following turbulence phenomenon. For a dense
subset of initial data in , the solutions tend to infinity in for every s\textgreater{}\frac
12 with super--polynomial growth on some sequence of times, while they go back
to their initial data on another sequence of times tending to infinity. This
transformation is defined by solving a general inverse spectral problem
involving singular values of a Hilbert--Schmidt Hankel operator and of its
shifted Hankel operator.Comment: arXiv admin note: substantial text overlap with arXiv:1402.171
Invariant tori for the cubic Szeg\"o equation
We continue the study of the following Hamiltonian equation on the Hardy
space of the circle, where denotes the
Szeg\"o projector. This equation can be seen as a toy model for totally non
dispersive evolution equations. In a previous work, we proved that this
equation admits a Lax pair, and that it is completely integrable. In this
paper, we construct the action-angle variables, which reduces the explicit
resolution of the equation to a diagonalisation problem. As a consequence, we
solve an inverse spectral problem for Hankel operators. Moreover, we establish
the stability of the corresponding invariant tori. Furthermore, from the
explicit formulae, we deduce the classification of orbitally stable and
unstable traveling waves
Multiple singular values of Hankel operators
The goal of this paper is to construct a nonlinear Fourier transformation on
the space of symbols of compact Hankel operators on the circle. This
transformation allows to solve a general inverse spectral problem involving
singular values of a compact Hankel operator, with arbitrary multiplicities.
The formulation of this result requires the introduction of the pair made with
a Hankel operator and its shifted Hankel operator. As an application, we prove
that the space of symbols of compact Hankel operators on the circle admits a
singular foliation made of tori of finite or infinite dimensions, on which the
flow of the cubic Szeg\"o equation acts. In particular, we infer that arbitrary
solutions of the cubic Szeg\"o equation on the circle with finite momentum are
almost periodic with values in H^{1/2}(S ^1).Comment: 80 page
Effective integrable dynamics for some nonlinear wave equation
We consider the following degenerate half wave equation on the one
dimensional torus We
show that, on a large time interval, the solution may be approximated by the
solution of a completely integrable system-- the cubic Szeg\"o equation. As a
consequence, we prove an instability result for large norms of solutions
of this wave equation.Comment: 19 page
Generic colourful tori and inverse spectral transform for Hankel operators
This paper explores the regularity properties of an inverse spectral
transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral
transform plays the role of action-angles variables for an integrable infinite
dimensional Hamiltonian system -- the cubic Szeg\"o equation. We investigate
the regularity of functions on the tori supporting the dynamics of this system,
in connection with some wave turbulence phenomenon, discovered in a previous
work and due to relative small gaps between the actions. We revisit this
phenomenon by proving that generic smooth functions and a G dense set
of irregular functions do coexist on the same torus. On the other hand, we
establish some uniform analytic regularity for tori corresponding to rapidly
decreasing actions which satisfy some specific property ruling out the
phenomenon of small gaps
Truncation of Multilinear Hankel operators
We extend to multilinear Hankel operators the fact that truncation of bounded
Hankel operators is bounded. We prove and use a continuity property of a kind
of bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces.Comment: 8 page
Harmonic functions on the real hyperbolic ball II Hardy and Lipschitz spaces
In this paper, we pursue the study of harmonic functions on the real
hyperbolic ball started by the second named author. Our focus here is on the
theory of Hardy, Hardy-Sobolev and Lipschitz spaces of these functions. We
prove here that these spaces admit Fefferman-Stein like characterizations in
terms of maximal and square functionals. We further prove that the hyperbolic
harmonic extension of Lipschitz functions on the boundary extend into Lipschitz
functions on the whole ball.Comment: 29 pages, 5 figures, LATEX file Authors partially supported by the
"European Commission" (TMR 1998-2001 Network Harmonic Analysis
Spectral inverse problems for compact Hankel operators
Given two arbitrary sequences and
of real numbers satisfying we prove that there exists a unique sequence
, real valued, such that the Hankel operators
and of symbols and respectively, are selfadjoint compact operators on
and have the sequences and respectively as non zero eigenvalues. Moreover, we give an explicit formula
for and we describe the kernel of and of in
terms of the sequences and . More
generally, given two arbitrary sequences and
of positive numbers satisfying
we describe
the set of sequences of complex numbers such that the
Hankel operators and are compact on and have sequences and
respectively as non zero singular values.Comment: 25 page
Truncations of multilinear Hankel operators
We extend to multilinear Hankel operators the fact that some truncations of
bounded Hankel operators are bounded. We prove and use a continuity property of
bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces
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