1,651 research outputs found
Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
We prove the most general theorem about spectral stability of multi-site
breathers in the discrete Klein-Gordon equation with a small coupling constant.
In the anti-continuum limit, multi-site breathers represent excited
oscillations at different sites of the lattice separated by a number of "holes"
(sites at rest). The theorem describes how the stability or instability of a
multi-site breather depends on the phase difference and distance between the
excited oscillators. Previously, only multi-site breathers with adjacent
excited sites were considered within the first-order perturbation theory. We
show that the stability of multi-site breathers with one-site holes change for
large-amplitude oscillations in soft nonlinear potentials. We also discover and
study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site
breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure
Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data
Let u be a solution to a quasi-linear Klein-Gordon equation in one-space
dimension, \partial\partial\partial\partial\partial , where P is a homogeneous polynomial of
degree three, and with smooth Cauchy data of size . It
is known that, under a suitable condition on the nonlinearity, the solution is
global-in-time for compactly supported Cauchy data. We prove in this paper that
the result holds even when data are not compactly supported but just decaying
as at infinity, combining the method of Klainerman
vector fields with a semiclassical normal forms method introduced by Delort.
Moreover, we get a one term asymptotic expansion for u when
Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation
We present some numerical findings concerning the nature of the blowup vs.
global existence dichotomy for the focusing cubic nonlinear Klein-Gordon
equation in three dimensions for radial data. The context of this study is
provided by the classical paper by Payne, Sattinger from 1975, as well as the
recent work by K. Nakanishi, and the second author arXiv:1005.4894.
Specifically, we numerically investigate the boundary of the forward scattering
region. At this point we do not have sufficient numerical evidence that might
indicate whether or not the boundary remains a smooth manifold for general
energies. In this updated version we include some fine-scale computations that
reveal more complicated structures than one might expect.Comment: 30 images. In this updated we include results that were obtained by
means of the CRAY XT5 supercomputer at the NICS, the National Institute of
Computational Sciences at Oakridge Labs, Tennessee, which is part of the
TeraGrid. Support by the NSF through TG-DMS110003 is gratefully acknowledge
Statistical mechanics of general discrete nonlinear Schr{\"o}dinger models: Localization transition and its relevance for Klein-Gordon lattices
We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical
mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS)
equation to a more general class of models, including higher dimensionalities
and nonlinearities of arbitrary degree. These extensions are physically
motivated by the desire to describe situations with an excitation threshold for
creation of localized excitations, as well as by recent work suggesting
non-cubic DNLS models to describe Bose-Einstein condensates in deep optical
lattices, taking into account the effective condensate dimensionality.
Considering ensembles of initial conditions with given values of the two
conserved quantities, norm and Hamiltonian, we calculate analytically the
boundary of the 'normal' Gibbsian regime corresponding to infinite temperature,
and perform numerical simulations to illuminate the nature of the localization
dynamics outside this regime for various cases. Furthermore, we show
quantitatively how this DNLS localization transition manifests itself for
small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled
anharmonic oscillators (in which energy is the only conserved quantity), and
determine conditions for existence of persistent energy localization over large
time scales.Comment: to be published in Physical Review
Translationally invariant nonlinear Schrodinger lattices
Persistence of stationary and traveling single-humped localized solutions in
the spatial discretizations of the nonlinear Schrodinger (NLS) equation is
addressed. The discrete NLS equation with the most general cubic polynomial
function is considered. Constraints on the nonlinear function are found from
the condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The discrete NLS
equation with such an exceptional nonlinear function is shown to have a
conserved momentum but admits no standard Hamiltonian structure. It is proved
that the reduction to the first-order difference map gives a sufficient
condition for existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped traveling
solutions. Other constraints on the nonlinear function are found from the
condition that the differential advance-delay equation for traveling solutions
admits a reduction to an integrable normal form given by a third-order
differential equation. This reduction also gives a necessary condition for
existence of single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete NLS equations
which generalizes the integrable Ablowitz--Ladik lattice.Comment: 24 pages, 4 figure
Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach
We study the persistence of the Gevrey class regularity of solutions to
nonlinear wave equations with real analytic nonlinearity. Specifically, it is
proven that the solution remains in a Gevrey class, with respect to some of its
spatial variables, during its whole life-span, provided the initial data is
from the same Gevrey class with respect to these spatial variables. In
addition, for the special Gevrey class of analytic functions, we find a lower
bound for the radius of the spatial analyticity of the solution that might
shrink either algebraically or exponentially, in time, depending on the
structure of the nonlinearity. The standard theory for the Gevrey class
regularity is employed; we also employ energy-like methods for a generalized
version of Gevrey classes based on the norm of Fourier transforms
(Wiener algebra). After careful comparisons, we observe an indication that the
approach provides a better lower bound for the radius of analyticity
of the solutions than the approach. We present our results in the case of
period boundary conditions, however, by employing exactly the same tools and
proofs one can obtain similar results for the nonlinear wave equations and the
nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain
domains and manifolds without physical boundaries, such as the whole space
, or on the sphere
Nonexistence of small, odd breathers for a class of nonlinear wave equations
In this note, we show that for a large class of nonlinear wave equations with
odd nonlinearities, any globally defined odd solution which is small in the
energy space decays to in the local energy norm. In particular, this result
shows nonexistence of small, odd breathers for some classical nonlinear Klein
Gordon equations such as the sine Gordon equation and and
models. It also partially answers a question of Soffer and Weinstein in
\cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in
dimension one
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