5,805 research outputs found
Scattering in the energy space for Boussinesq equations
In this note we show that all small solutions in the energy space of the
generalized 1D Boussinesq equation must decay to zero as time tends to
infinity, strongly on slightly proper subsets of the space-time light cone. Our
result does not require any assumption on the power of the nonlinearity,
working even for the supercritical range of scattering. No parity assumption on
the initial data is needed
Scalar field breathers on anti-de Sitter background
We study spatially localized, time-periodic solutions (breathers) of scalar
field theories with various self-interacting potentials on Anti-de Sitter (AdS)
spacetimes in dimensions. A detailed numerical study of spherically
symmetric configurations in dimensions is carried out, revealing a rich
and complex structure of the phase-space (bifurcations, resonances). Scalar
breather solutions form one-parameter families parametrized by their amplitude,
, while their frequency, , is a
function of the amplitude. The scalar breathers on AdS we find have a small
amplitude limit, tending to the eigenfunctions of the linear Klein-Gordon
operator on AdS. Importantly most of these breathers appear to be generically
stable under time evolution.Comment: 30 pages, 22 figure
The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields
We study the initial value problem for two fundamental theories of gravity,
that is, Einstein's field equations of general relativity and the
(fourth-order) field equations of f(R) modified gravity. For both of these
physical theories, we investigate the global dynamics of a self-gravitating
massive matter field when an initial data set is prescribed on an
asymptotically flat and spacelike hypersurface, provided these data are
sufficiently close to data in Minkowski spacetime. Under such conditions, we
thus establish the global nonlinear stability of Minkowski spacetime in
presence of massive matter. In addition, we provide a rigorous mathematical
validation of the f(R) theory based on analyzing a singular limit problem, when
the function f(R) arising in the generalized Hilbert-Einstein functional
approaches the scalar curvature function R of the standard Hilbert-Einstein
functional. In this limit we prove that f(R) Cauchy developments converge to
Einstein's Cauchy developments in the regime close to Minkowski space. Our
proofs rely on a new strategy, introduced here and referred to as the
Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of
the Hyperboloidal Foliation Method (HFM) which we used earlier for the
Einstein-massive field system but for a restricted class of initial data. Here,
the data are solely assumed to satisfy an asymptotic flatness condition and be
small in a weighted energy norm. These results for matter spacetimes provide a
significant extension to the existing stability theory for vacuum spacetimes,
developed by Christodoulou and Klainerman and revisited by Lindblad and
Rodnianski.Comment: 127 pages. Selected chapters from a boo
The maximum principle and sign changing solutions of the hyperbolic equation with the Higgs potential
In this article we discuss the maximum principle for the linear equation and
the sign changing solutions of the semilinear equation with the Higgs
potential. Numerical simulations indicate that the bubbles for the semilinear
Klein-Gordon equation in the de Sitter spacetime are created and apparently
exist for all times
KAM for the Klein Gordon equation on
Recently the KAM theory has been extended to multidimensional PDEs.
Nevertheless all these recent results concern PDEs on the torus, essentially
because in that case the corresponding linear PDE is diagonalized in the
Fourier basis and the structure of the resonant sets is quite simple. In the
present paper, we consider an important physical example that do not fit in
this context: the Klein Gordon equation on . Our abstract KAM
theorem also allow to prove the reducibility of the corresponding linear
operator with time quasiperiodic potentials.Comment: arXiv admin note: substantial text overlap with arXiv:1410.808
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