864 research outputs found
Perturbed Three Vortex Dynamics
It is well known that the dynamics of three point vortices moving in an ideal
fluid in the plane can be expressed in Hamiltonian form, where the resulting
equations of motion are completely integrable in the sense of Liouville and
Arnold. The focus of this investigation is on the persistence of regular
behavior (especially periodic motion) associated to completely integrable
systems for certain (admissible) kinds of Hamiltonian perturbations of the
three vortex system in a plane. After a brief survey of the dynamics of the
integrable planar three vortex system, it is shown that the admissible class of
perturbed systems is broad enough to include three vortices in a half-plane,
three coaxial slender vortex rings in three-space, and `restricted' four vortex
dynamics in a plane. Included are two basic categories of results for
admissible perturbations: (i) general theorems for the persistence of invariant
tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff
type arguments; and (ii) more specific and quantitative conclusions of a
classical perturbation theory nature guaranteeing the existence of periodic
orbits of the perturbed system close to cycles of the unperturbed system, which
occur in abundance near centers. In addition, several numerical simulations are
provided to illustrate the validity of the theorems as well as indicating their
limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic
Pulses and Snakes in Ginzburg--Landau Equation
Using a variational formulation for partial differential equations (PDEs)
combined with numerical simulations on ordinary differential equations (ODEs),
we find two categories (pulses and snakes) of dissipative solitons, and analyze
the dependence of both their shape and stability on the physical parameters of
the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular
solitary waves investigated in numerous integrable and non-integrable systems
over the last three decades, these dissipative solitons are not stationary in
time. Rather, they are spatially confined pulse-type structures whose envelopes
exhibit complicated temporal dynamics. Numerical simulations reveal very
interesting bifurcations sequences as the parameters of the CGLE are varied.
Our predictions on the variation of the soliton amplitude, width, position,
speed and phase of the solutions using the variational formulation agree with
simulation results.Comment: 30 pages, 14 figure
Manipulation of the dynamics of many-body systems via quantum control methods
We investigate how dynamical decoupling methods may be used to manipulate the
time evolution of quantum many-body systems. These methods consist of sequences
of external control operations designed to induce a desired dynamics. The
systems considered for the analysis are one-dimensional spin-1/2 models, which,
according to the parameters of the Hamiltonian, may be in the integrable or
non-integrable limits, and in the gapped or gapless phases. We show that an
appropriate control sequence may lead a chaotic chain to evolve as an
integrable chain and a system in the gapless phase to behave as a system in the
gapped phase. A key ingredient for the control schemes developed here is the
possibility to use, in the same sequence, different time intervals between
control operations.Comment: 10 pages, 3 figure
Coherent oscillations and incoherent tunnelling in one - dimensional asymmetric double - well potential
For a model 1d asymmetric double-well potential we calculated so-called
survival probability (i.e. the probability for a particle initially localised
in one well to remain there). We use a semiclassical (WKB) solution of
Schroedinger equation. It is shown that behaviour essentially depends on
transition probability, and on dimensionless parameter which is a ratio of
characteristic frequencies for low energy non-linear in-well oscillations and
inter wells tunnelling. For the potential describing a finite motion
(double-well) one has always a regular behaviour. For the small value of the
parameter there is well defined resonance pairs of levels and the survival
probability has coherent oscillations related to resonance splitting. However
for the large value of the parameter no oscillations at all for the survival
probability, and there is almost an exponential decay with the characteristic
time determined by Fermi golden rule. In this case one may not restrict oneself
to only resonance pair levels. The number of perturbed by tunnelling levels
grows proportionally to the value of this parameter (by other words instead of
isolated pairs there appear the resonance regions containing the sets of
strongly coupled levels). In the region of intermediate values of the parameter
one has a crossover between both limiting cases, namely the exponential decay
with subsequent long period recurrent behaviour.Comment: 19 pages, 7 figures, Revtex, revised version. Accepted to Phys. Rev.
The Hess-Appelrot system and its nonholonomic analogs
This paper is concerned with the nonholonomic Suslov problem and its
generalization proposed by Chaplygin. The issue of the existence of an
invariant measure with singular density (having singularities at some points of
phase space) is discussed
Helmholtz bright and boundary solitons
We report, for the first time, exact analytical boundary solitons of a generalized cubic-quintic Non-Linear Helmholtz (NLH) equation. These solutions have a linked-plateau topology that is distinct from conventional dark soliton solutions; their amplitude and intensity distributions are spatially delocalized and connect regions of finite and zero wave-field disturbances (suggesting also the classification as 'edge solitons'). Extensive numerical simulations compare the stability properties of recently-reported Helmholtz bright solitons, for this type of polynomial non-linearity, to those of the new boundary solitons. The latter are found to possess a remarkable stability characteristic, exhibiting robustness against perturbations that would otherwise lead to the destabilizing of their bright-soliton counterpart
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