145 research outputs found
Spiral anchoring in anisotropic media with multiple inhomogeneities: a dynamical system approach
Various PDE models have been suggested in order to explain and predict the
dynamics of spiral waves in excitable media. In two landmark papers, Barkley
noticed that some of the behaviour could be explained by the inherent Euclidean
symmetry of these models. LeBlanc and Wulff then introduced forced Euclidean
symmetry-breaking (FESB) to the analysis, in the form of individual
translational symmetry-breaking (TSB) perturbations and rotational
symmetry-breaking (RSB) perturbations; in either case, it is shown that spiral
anchoring is a direct consequence of the FESB.
In this article, we provide a characterization of spiral anchoring when two
perturbations, a TSB term and a RSB term, are combined, where the TSB term is
centered at the origin and the RSB term preserves rotations by multiples of
, where is an integer. When
(such as in a modified bidomain model), it is shown that spirals
anchor at the origin, but when (such as in a planar
reaction-diffusion-advection system), spirals generically anchor away from the
origin.Comment: Revised versio
Coarse-grained dynamics of an activity bump in a neural field model
We study a stochastic nonlocal PDE, arising in the context of modelling
spatially distributed neural activity, which is capable of sustaining
stationary and moving spatially-localized ``activity bumps''. This system is
known to undergo a pitchfork bifurcation in bump speed as a parameter (the
strength of adaptation) is changed; yet increasing the noise intensity
effectively slowed the motion of the bump. Here we revisit the system from the
point of view of describing the high-dimensional stochastic dynamics in terms
of the effective dynamics of a single scalar "coarse" variable. We show that
such a reduced description in the form of an effective Langevin equation
characterized by a double-well potential is quantitatively successful. The
effective potential can be extracted using short, appropriately-initialized
bursts of direct simulation. We demonstrate this approach in terms of (a) an
experience-based "intelligent" choice of the coarse observable and (b) an
observable obtained through data-mining direct simulation results, using a
diffusion map approach.Comment: Corrected aknowledgement
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
In a companion paper, we established nonlinear stability with detailed
diffusive rates of decay of spectrally stable periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. Here, we determine
time-asymptotic behavior under such perturbations, showing that solutions
consist to leading order of a modulation whose parameter evolution is governed
by an associated Whitham averaged equation
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Coupled-mode systems are used in physical literature to simplify the
nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic
potential and to approximate localized solutions called gap solitons by
analytical expressions involving hyperbolic functions. We justify the use of
the one-dimensional stationary coupled-mode system for a relevant elliptic
problem by employing the method of Lyapunov--Schmidt reductions in Fourier
space. In particular, existence of periodic/anti-periodic and decaying
solutions is proved and the error terms are controlled in suitable norms. The
use of multi-dimensional stationary coupled-mode systems is justified for
analysis of bifurcations of periodic/anti-periodic solutions in a small
multi-dimensional periodic potential.Comment: 18 pages, no figure
Stability of pulses on optical fibers with phase-sensitive amplifiers
Pulse stability is crucial to the effective propagation of information in a soliton-based optical communication system. It is shown in this paper that pulses in optical fibers, for which attenuation is compensated by phase-sensitive amplifiers, are stable over a large range of parameter values. A fourth-order nonlinear diffusion model due to Kath and co-workers is used. The stability proof invokes a number of mathematical techniques, including the Evans function and Grillakis' functional analytic approach
Discrete embedded solitons
We address the existence and properties of discrete embedded solitons (ESs),
i.e., localized waves existing inside the phonon band in a nonlinear
dynamical-lattice model. The model describes a one-dimensional array of optical
waveguides with both the quadratic (second-harmonic generation) and cubic
nonlinearities. A rich family of ESs was previously known in the continuum
limit of the model. First, a simple motivating problem is considered, in which
the cubic nonlinearity acts in a single waveguide. An explicit solution is
constructed asymptotically in the large-wavenumber limit. The general problem
is then shown to be equivalent to the existence of a homoclinic orbit in a
four-dimensional reversible map. From properties of such maps, it is shown that
(unlike ordinary gap solitons), discrete ESs have the same codimension as their
continuum counterparts. A specific numerical method is developed to compute
homoclinic solutions of the map, that are symmetric under a specific reversing
transformation. Existence is then studied in the full parameter space of the
problem. Numerical results agree with the asymptotic results in the appropriate
limit and suggest that the discrete ESs may be semi-stable as in the continuous
case.Comment: A revtex4 text file and 51 eps figure files. To appear in
Nonlinearit
Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback
The onset of pulse propagation is studied in a reaction-diffusion (RD) model
with control by augmented transmission capability that is provided either along
nonlocal spatial coupling or by time-delayed feedback. We show that traveling
pulses occur primarily as solutions to the RD equations while augmented
transmission changes excitability. For certain ranges of the parameter
settings, defined as weak susceptibility and moderate control, respectively,
the hybrid model can be mapped to the original RD model. This results in an
effective change of RD parameters controlled by augmented transmission. Outside
moderate control parameter settings new patterns are obtained, for example
step-wise propagation due to delay-induced oscillations. Augmented transmission
constitutes a signaling system complementary to the classical RD mechanism of
pattern formation. Our hybrid model combines the two major signaling systems in
the brain, namely volume transmission and synaptic transmission. Our results
provide insights into the spread and control of pathological pulses in the
brain
Vortices in a Bose-Einstein condensate confined by an optical lattice
We investigate the dynamics of vortices in repulsive Bose-Einstein
condensates in the presence of an optical lattice (OL) and a parabolic magnetic
trap. The dynamics is sensitive to the phase of the OL potential relative to
the magnetic trap, and depends less on the OL strength. For the cosinusoidal OL
potential, a local minimum is generated at the trap's center, creating a stable
equilibrium for the vortex, while in the case of the sinusoidal potential, the
vortex is expelled from the center, demonstrating spiral motion. Cases where
the vortex is created far from the trap's center are also studied, revealing
slow outward-spiraling drift. Numerical results are explained in an analytical
form by means of a variational approximation. Finally, motivated by a discrete
model (which is tantamount to the case of the strong OL lattice), we present a
novel type of vortex consisting of two pairs of anti-phase solitons.Comment: 10 pages, 6 figure
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