164 research outputs found
On the Existence of Localized Excitations in Nonlinear Hamiltonian Lattices
We consider time-periodic nonlinear localized excitations (NLEs) on
one-dimensional translationally invariant Hamiltonian lattices with arbitrary
finite interaction range and arbitrary finite number of degrees of freedom per
unit cell. We analyse a mapping of the Fourier coefficients of the NLE
solution. NLEs correspond to homoclinic points in the phase space of this map.
Using dimensionality properties of separatrix manifolds of the mapping we show
the persistence of NLE solutions under perturbations of the system, provided
NLEs exist for the given system. For a class of nonintegrable Fermi-Pasta-Ulam
chains we rigorously prove the existence of NLE solutions.Comment: 13 pages, LaTeX, 2 figures will be mailed upon request (Phys. Rev. E,
in press
Generalized Schr\"odinger-Newton system in dimension : critical case
In this paper we study a system which is equivalent to a nonlocal version of
the well known Brezis Nirenberg problem. The difficulties related with the lack
of compactness are here emphasized by the nonlocal nature of the critical
nonlinear term. We prove existence and nonexistence results of positive
solutions when and existence of solutions in both the resonance and the
nonresonance case for higher dimensions.Comment: 18 pages, typos fixed, some minor revision
Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction
We propose a unified approach to nonlinear modal analysis in dissipative
oscillatory systems. This approach eliminates conflicting definitions, covers
both autonomous and time-dependent systems, and provides exact mathematical
existence, uniqueness and robustness results. In this setting, a nonlinear
normal mode (NNM) is a set filled with small-amplitude recurrent motions: a
fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In
contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a
NNM, serving as the smoothest nonlinear continuation of a spectral subspace of
the linearized system along the NNM. The existence and uniqueness of SSMs turns
out to depend on a spectral quotient computed from the real part of the
spectrum of the linearized system. This quotient may well be large even for
small dissipation, thus the inclusion of damping is essential for firm
conclusions about NNMs, SSMs and the reduced-order models they yield.Comment: To appear in Nonlinear Dynamic
Stability Properties of the Riemann Ellipsoids
We study the ellipticity and the ``Nekhoroshev stability'' (stability
properties for finite, but very long, time scales) of the Riemann ellipsoids.
We provide numerical evidence that the regions of ellipticity of the ellipsoids
of types II and III are larger than those found by Chandrasekhar in the 60's
and that all Riemann ellipsoids, except a finite number of codimension one
subfamilies, are Nekhoroshev--stable. We base our analysis on a Hamiltonian
formulation of the problem on a covering space, using recent results from
Hamiltonian perturbation theory.Comment: 29 pages, 6 figure
Linear elliptic system with nonlinear boundary conditions without Landesman-Lazer conditions
The boundary value problem is examined for the system of elliptic equations
of from where is positive
semidefinite matrix on and It is assumed that
is a bounded function which may vanish
at infinity. The proofs are based on Leray-Schauder degree methods
Non resonance conditions for radial solutions of non linear Neumann elliptic problems on annuli
Andrea Sfecci, "Non resonance conditions for radial solutions of non linear Neumann elliptic problems on annuli", in: Rendiconti dellâIstituto di Matematica dellâUniversitĂ di Trieste. An International Journal of Mathematics, 46 (2014), pp.255-270An existence result to some nonlinear Neumann elliptic problems defined on balls has been provided recently by the author in [21]. We investigate, in this paper, the possibility of extending such a result to annuli
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Local energy decay for scalar fields on time dependent non-trapping backgrounds
We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a âtwo pointâ local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate
Renormalization Group and the Melnikov Problem for PDE's
We give a new proof of persistence of quasi-periodic, low dimensional
elliptic tori in infinite dimensional systems. The proof is based on a
renormalization group iteration that was developed recently in [BGK] to address
the standard KAM problem, namely, persistence of invariant tori of maximal
dimension in finite dimensional, near integrable systems. Our result covers
situations in which the so called normal frequencies are multiple. In
particular, it provides a new proof of the existence of small-amplitude,
quasi-periodic solutions of nonlinear wave equations with periodic boundary
conditions.Comment: 44 pages, plain Te
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