686 research outputs found
Turing pattern outside of the Turing domain
There are two simple solutions to reaction-diffusion systems with limit-cycle reaction kinetics, producing oscillatory behaviour. The reaction parameter gives rise to a ‘space-invariant’ solution, and versus the ratio of the diffusion coefficients gives rise to a ‘time-invariant’ solution. We consider the case where both solution types may be possible. This leads to a refinement of the Turing model of pattern formation. We add convection to the system and investigate its effect. More complex solutions arise that appear to combine the two simple solutions. The convective system sheds light on the underlying behaviour of the diffusive system
Limit cycles in the presence of convection, a first order analysis
We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads pattern outwards from the source. Convection adds instability to the reaction-diffusion system. We see the result of the instability in a readiness to create pattern. In the case of strong convection, we consider that the first-order approximation may be valid for some aspects of the solution behaviour. We employ the method of Riemann invariants and rescaling to transform the reduced system into one invariant under parameter change. We carry out numerical experiments to test our analysis. We find that most aspects of the solution do not comply with this, but we find one significant characteristic which is approximately first order. We consider the correspondence of the Partial Differential Equation with the Ordinary Differential Equation along rays from the initiation point in the transformed system. This yields an understanding of the behaviour
Discrete breathers in classical spin lattices
Discrete breathers (nonlinear localised modes) have been shown to exist in
various nonlinear Hamiltonian lattice systems. In the present paper we study
the dynamics of classical spins interacting via Heisenberg exchange on spatial
-dimensional lattices (with and without the presence of single-ion
anisotropy). We show that discrete breathers exist for cases when the continuum
theory does not allow for their presence (easy-axis ferromagnets with
anisotropic exchange and easy-plane ferromagnets). We prove the existence of
localised excitations using the implicit function theorem and obtain necessary
conditions for their existence. The most interesting case is the easy-plane one
which yields excitations with locally tilted magnetisation. There is no
continuum analogue for such a solution and there exists an energy threshold for
it, which we have estimated analytically. We support our analytical results
with numerical high-precision computations, including also a stability analysis
for the excitations.Comment: 15 pages, 12 figure
Nonlinear Fano resonance and bistable wave transmission
We consider a discrete model that describes a linear chain of particles
coupled to a single-site defect with instantaneous Kerr nonlinearity. We show
that this model can be regarded as a nonlinear generalization of the familiar
Fano-Anderson model, and it can generate the amplitude depended bistable
resonant transmission or reflection. We identify these effects as the nonlinear
Fano resonance, and study its properties for continuous waves and pulses.Comment: 9 pages, 14 figure, submitted to Phys. Rev.
Limit cycles in the presence of convection, a travelling wave analysis
We consider a diffusion model with limit cycle reaction functions, in the
presence of convection. We select a set of functions derived from a realistic
reaction model: the Schnakenberg equations. This resultant form is
unsymmetrical. We find a transformation which maps the irregular equations into
model form. Next we transform the dependent variables into polar form. From
here, a travelling wave analysis is performed on the radial variable. Results
are complex, but we make some simple estimates.
We carry out numerical experiments to test our analysis. An initial `knock'
starts the propagation of pattern. The speed of the travelling wave is not
quite as expected. We investigate further. The system demonstrates distinctly
different behaviour to the left and the right. We explain how this phenomenon
occurs by examining the underlying behaviour.Comment: 20 pages, 5 figure
Incremental expansions for Hubbard-Peierls systems
The ground state energies of infinite half-filled Hubbard-Peierls chains are
investigated combining incremental expansion with exact diagonalization of
finite chain segments. The ground state energy of equidistant infinite Hubbard
(Heisenberg) chains is calculated with a relative error of less than for all values of using diagonalizations of 12-site (20-site)
chain segm ents. For dimerized chains the dimerization order parameter as a
function of the onsite repulsion interaction has a maximum at nonzero
values of , if the electron-phonon coupling is lower than a critical
value . The critical value is found with high accuracy to be
. For smaller values of the position of the maximum of is
approximately , and rapidly tends to zero as approaches from
below. We show how our method can be applied to calculate breathers for the
problem of phonon dynamics in Hubbard-Peierls systems.Comment: 4 Pages, 3 Figures, REVTE
Observation of breathers in Josephson ladders
We report on the observation of spatially-localized excitations in a ladder
of small Josephson junctions. The excitations are whirling states which persist
under a spatially-homogeneous force due to the bias current. These states of
the ladder are visualized using a low temperature scanning laser microscopy. We
also compute breather solutions with high accuracy in corresponding model
equations. The stability analysis of these solutions is used to interpret the
measured patterns in the I-V characteristics
Vortex and translational currents due to broken time-space symmetries
We consider the classical dynamics of a particle in a -dimensional
space-periodic potential under the influence of time-periodic external fields
with zero mean. We perform a general time-space symmetry analysis and identify
conditions, when the particle will generate a nonzero averaged translational
and vortex currents. We perform computational studies of the equations of
motion and of corresponding Fokker-Planck equations, which confirm the symmetry
predictions. We address the experimentally important issue of current control.
Cold atoms in optical potentials and magnetic traps are among possible
candidates to observe these findings experimentally.Comment: 4 pages, 2 figure
Tunneling of quantum rotobreathers
We analyze the quantum properties of a system consisting of two nonlinearly
coupled pendula. This non-integrable system exhibits two different symmetries:
a permutational symmetry (permutation of the pendula) and another one related
to the reversal of the total momentum of the system. Each of these symmetries
is responsible for the existence of two kinds of quasi-degenerated states. At
sufficiently high energy, pairs of symmetry-related states glue together to
form quadruplets. We show that, starting from the anti-continuous limit,
particular quadruplets allow us to construct quantum states whose properties
are very similar to those of classical rotobreathers. By diagonalizing
numerically the quantum Hamiltonian, we investigate their properties and show
that such states are able to store the main part of the total energy on one of
the pendula. Contrary to the classical situation, the coupling between pendula
necessarily introduces a periodic exchange of energy between them with a
frequency which is proportional to the energy splitting between
quasi-degenerated states related to the permutation symmetry. This splitting
may remain very small as the coupling strength increases and is a decreasing
function of the pair energy. The energy may be therefore stored in one pendulum
during a time period very long as compared to the inverse of the internal
rotobreather frequency.Comment: 20 pages, 11 figures, REVTeX4 styl
Tunable Pinning of Burst-Waves in Extended Systems with Discrete Sources
We study the dynamics of waves in a system of diffusively coupled discrete
nonlinear sources. We show that the system exhibits burst waves which are
periodic in a traveling-wave reference frame. We demonstrate that the burst
waves are pinned if the diffusive coupling is below a critical value. When the
coupling crosses the critical value the system undergoes a depinning
instability via a saddle-node bifurcation, and the wave begins to move. We
obtain the universal scaling for the mean wave velocity just above threshold.Comment: 4 pages, 5 figures, revte
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