532 research outputs found
Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation
We construct center-stable and center-unstable manifolds, as well as stable
and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing
energy sub-critical nonlinearity, associated with a family of solitary waves
which is generated from any radial stationary solution by the action of all
Lorentz transforms and spatial translations. The construction is based on the
graph transform (or Hadamard) approach, which requires less spectral
information on the linearized operator, and less decay of the nonlinearity,
than the Lyapunov-Perron method employed previously in this context. The only
assumption on the stationary solution is that the kernel of the linearized
operator is spanned by its spatial derivatives, which is known to hold for the
ground states. The main novelty of this paper lies with the fact that the graph
transform method is carried out in the presence of modulation parameters
corresponding to the symmetries.Comment: 38 page
Critical velocity in kink solutions of the sine-Gordon equation
The goal of this work is to present a way to deduce the value of the critical velocity observed in soliton-like solutions of a perturbed version of the sine-Gordon equation. To do so, an ODE system is obtained from the perturbed sine-Gordon equation using a variational approach; the resulting Hamiltonian system is then studied. From that, a Melnikov integral formula for the critical velocity is deduced via an energy balance reasoning. Finally, the problem is approached from a geometrical point of view that allows for an interpretation of the previous results in terms of intersections of invariant manifolds of periodic orbits
Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with
large radial data in the energy space. This equation admits a unique positive
stationary solution, called the ground state. In 1975, Payne and Sattinger
showed that solutions with energy strictly below that of the ground state are
divided into two classes, depending on a suitable functional: If it is
negative, then one has finite time blowup, if it is nonnegative, global
existence; moreover, these sets are invariant under the flow. Recently,
Ibrahim, Masmoudi and the first author improved this result by establishing
scattering to zero in the global existence case by means of a variant of the
Kenig-Merle method. In this paper we go slightly beyond the ground state energy
and give a complete description of the evolution. For example, in a small
neighborhood of the ground states one encounters the following trichotomy: on
one side of a center-stable manifold one has finite-time blowup, on the other
side scattering to zero, and on the manifold itself one has scattering to the
ground state, all for positive time. In total, the class of initial data is
divided into nine disjoint nonempty sets, each displaying different asymptotic
behavior, which includes solutions blowing up in one time direction and
scattering to zero on the other, and also, the analogue of those found by
Duyckaerts and Merle for the energy critical wave and Schr\"odinger equations,
exactly with the ground state energy. The main technical ingredient is a
"one-pass" theorem which excludes the existence of "almost homoclinic" orbits
between the ground states.Comment: 34 pages, minor correction
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
Global dynamics above the ground state energy for the one-dimensional NLKG equation
In this paper we obtain a global characterization of the dynamics of even
solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the
line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds
that of the ground state only sightly. The method is the same as in the
three-dimensional case arXiv:1005.4894, the major difference being in the
construction of the center-stable manifold. The difficulty there lies with the
weak dispersive decay of 1-dimensional NLKG. In order to address this specific
issue, we establish local dispersive estimates for the perturbed linear
Klein-Gordon equation, similar to those of Mizumachi arXiv:math/0605031. The
essential ingredient for the latter class of estimates is the absence of a
threshold resonance of the linearized operator
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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