286 research outputs found
On the Dynamics of solitons in the nonlinear Schroedinger equation
We study the behavior of the soliton solutions of the equation
i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi}
where W_{{\epsilon}}' is a suitable nonlinear term which is singular for
{\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence,
shape, stability and dynamics of the soliton. The main result of this paper
(Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches
the orbit of a classical particle in a potential V(x).Comment: 29 page
Hylomorphic solitons in the nonlinear Klein-Gordon equation
Roughly speaking a solitary wave is a solution of a field equation whose
energy travels as a localised packet and which preserves this localisation in
time. A soliton is a solitary wave which exhibits some strong form of stability
so that it has a particle-like behaviour. In this paper we show a new mechanism
which might produce solitary waves and solitons for a large class of equations,
such as the nonlinear Klein-Gordon equation. We show that the existence of
these kind of solitons, that we have called \emph{hylomorphic} solitons,
depends on a suitable energy/charge ratio. We show a variational method that
allows to prove the existence of hylomorphic solitons and that turns out to be
very useful for numerical applications. Moreover we introduce some classes of
nonlinearities which admit hylomorphic solitons of different shapes and with
different relations between charge, energy and frequency.Comment: 23 page
Compression and diffusion: a joint approach to detect complexity
The adoption of the Kolmogorov-Sinai (KS) entropy is becoming a popular
research tool among physicists, especially when applied to a dynamical system
fitting the conditions of validity of the Pesin theorem. The study of time
series that are a manifestation of system dynamics whose rules are either
unknown or too complex for a mathematical treatment, is still a challenge since
the KS entropy is not computable, in general, in that case. Here we present a
plan of action based on the joint action of two procedures, both related to the
KS entropy, but compatible with computer implementation through fast and
efficient programs. The former procedure, called Compression Algorithm
Sensitive To Regularity (CASToRe), establishes the amount of order by the
numerical evaluation of algorithmic compressibility. The latter, called Complex
Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA),
establishes the complexity degree through the numerical evaluation of the
strength of an anomalous effect. This is the departure, of the diffusion
process generated by the observed fluctuations, from ordinary Brownian motion.
The CASSANDRA algorithm shares with CASToRe a connection with the Kolmogorov
complexity. This makes both algorithms especially suitable to study the
transition from dynamics to thermodynamics, and the case of non-stationary time
series as well. The benefit of the joint action of these two methods is proven
by the analysis of artificial sequences with the same main properties as the
real time series to which the joint use of these two methods will be applied in
future research work.Comment: 27 pages, 9 figure
Displacement energy of unit disk cotangent bundles
We give an upper bound of a Hamiltonian displacement energy of a unit disk
cotangent bundle in a cotangent bundle , when the base manifold
is an open Riemannian manifold. Our main result is that the displacement
energy is not greater than , where is the inner radius of ,
and is a dimensional constant. As an immediate application, we study
symplectic embedding problems of unit disk cotangent bundles. Moreover,
combined with results in symplectic geometry, our main result shows the
existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math
Zei
Convex domains of Finsler and Riemannian manifolds
A detailed study of the notions of convexity for a hypersurface in a Finsler
manifold is carried out. In particular, the infinitesimal and local notions of
convexity are shown to be equivalent. Our approach differs from Bishop's one in
his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the
Riemannian case. Ours not only can be extended to the Finsler setting but it
also reduces the typical requirements of differentiability for the metric and
it yields consequences on the multiplicity of connecting geodesics in the
convex domain defined by the hypersurface.Comment: 22 pages, AMSLaTex. Typos corrected, references update
Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with
smooth n-1 dimensional boundary. We search the positive solutions of the
singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann
boundary conditions or for the singularly perturbed Klein Gordon Maxwell system
with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that
stable critical points of the mean curvature of the boundary generates
solutions when the perturbation parameter is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1410.884
Calibrated Surfaces and Supersymmetric Wilson Loops
We study the dual gravity description of supersymmetric Wilson loops whose
expectation value is unity. They are described by calibrated surfaces that end
on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect
to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5.
The regularized area of these surfaces vanishes, in agreement with field theory
non-renormalization theorems for the corresponding operators.Comment: 28 pages, 2 figure
The tale of two centres
We study motion in the field of two fixed centres described by a family of
Einstein-dilaton-Maxwell theories. Transitions between regular and chaotic
motion are observed as the dilaton coupling is varied.Comment: 20 pages, RevTeX, 7 figures included, TeX format change
Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation
Einstein's relation E=Mc^2 between the energy E and the mass M is the
cornerstone of the relativity theory. This relation is often derived in a
context of the relativistic theory for closed systems which do not accelerate.
By contrast, Newtonian approach to the mass is based on an accelerated motion.
We study here a particular neoclassical field model of a particle governed by a
nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the
nonlinear KG equation and its energy density concentrate at a trajectory, then
this trajectory and the energy must satisfy the relativistic version of
Newton's law with the mass satisfying Einstein's relation. Therefore the
internal energy of a localized wave affects its acceleration in an external
field as the inertial mass does in Newtonian mechanics. We demonstrate that the
"concentration" assumptions hold for a wide class of rectilinear accelerating
motions
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