764 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
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
Threshold phenomenon for the quintic wave equation in three dimensions
For the critical focusing wave equation on in the
radial case, we establish the role of the "center stable" manifold
constructed in \cite{KS} near the ground state as a threshold between
type I blowup and scattering to zero, establishing a conjecture going back to
numerical work by Bizo\'n, Chmaj, Tabor. The underlying topology is stronger
than the energy norm
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
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