221 research outputs found
Boundary effects on the dynamics of chains of coupled oscillators
We study the dynamics of a chain of coupled particles subjected to a
restoring force (Klein-Gordon lattice) in the cases of either periodic or
Dirichlet boundary conditions. Precisely, we prove that, when the initial data
are of small amplitude and have long wavelength, the main part of the solution
is interpolated by a solution of the nonlinear Schr\"odinger equation, which in
turn has the property that its Fourier coefficients decay exponentially. The
first order correction to the solution has Fourier coefficients that decay
exponentially in the periodic case, but only as a power in the Dirichlet case.
In particular our result allows one to explain the numerical computations of
the paper \cite{BMP07}
Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein--Gordon lattices
We construct small amplitude breathers in 1D and 2D Klein--Gordon infinite
lattices. We also show that the breathers are well approximated by the ground
state of the nonlinear Schroedinger equation. The result is obtained by
exploiting the relation between the Klein Gordon lattice and the discrete Non
Linear Schroedinger lattice. The proof is based on a Lyapunov-Schmidt
decomposition and continuum approximation techniques introduced in [7],
actually using its main result as an important lemma
Quasi periodic breathers in Hamiltonian lattices with symmetries
We prove existence of quasiperiodic breathers in Hamiltonian lattices of weakly coupled oscillators having some integrals of motion independent of the Hamiltonian. The proof is obtained by constructing quasiperiodic breathers in the anticontinuoum limit and using a recent theorem by N.N. Nekhoroshev [8] as extended in [5] to continue them to the coupled case. Applications to several models are given
Invariant tori for commuting Hamiltonian PDEs
We generalize to some PDEs a theorem by Nekhoroshev on the persistence of
invariant tori in Hamiltonian systems with integrals of motion and
degrees of freedom, . The result we get ensures the persistence of an
-parameter family of -dimensional invariant tori. The parameters belong
to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition
and on the standard implicit function theorem. Some of the persistent tori are
resonant. We also give an application to the nonlinear wave equation with
periodic boundary conditions on a segment and to a system of coupled beam
equations. In the first case we construct 2 dimensional tori, while in the
second case we construct 3 dimensional tori
Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators
We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It
is well known that if the coupling is weak enough then the system admits
families of periodic solutions exponentially localized in space (breathers). In
this paper we prove asymptotic stability in energy space of such solutions. The
proof is based on two steps: first we use canonical perturbation theory to put
the system in a suitable normal form in a neighborhood of the breather, second
we use dispersion in order to prove asymptotic stability. The main limitation
of the result rests in the fact that the nonlinear part of the on site
potential is required to have a zero of order 8 at the origin. From a technical
point of view the theory differs from that developed for Hamiltonian PDEs due
to the fact that the breather is not a relative equilibrium of the system
Asymptotic stability of solitons
We consider a ground state (soliton) of NLS in dimension three. We prove
that if the soliton is orbitally stable, then it is also
asymptotically stable. The main assumptions are transversal
nondegeneracy of the manifold of the ground states, linear dispersion
(in the form of Strichartz estimates) and nonlinear Fermi Golden
Rule
Growth of Sobolev norms for abstract linear Schrodinger equations
We prove an abstract theorem giving a (t)ϵ bound (for all ϵ > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i Ψ = H0ψ + V(t)ψ as t → ∞. The abstract theorem is applied to several cases, including the cases where (i) H0 is the Laplace operator on a Zoll manifold and V (t) a pseudodifferential operator of order smaller than 2; (ii) H0 is the (resonant or nonresonant) harmonic oscillator in Rd and V (t) a pseudodifferential operator of order smaller than that of H0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17]
Normal Forms for Semilinear Quantum Harmonic Oscillators
We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2}
+M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where is
a Hermite multiplier and a smooth function globally of order 3 at least. We
prove that such a Hamiltonian equation admits, in a neighborhood of the origin,
a Birkhoff normal form at any order and that, under generic conditions on
related to the non resonance of the linear part, this normal form is integrable
when and gives rise to simple (in particular bounded) dynamics when
. As a consequence we prove the almost global existence for solutions
of the above equation with small Cauchy data. Furthermore we control the high
Sobolev norms of these solutions
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