68,380 research outputs found
Existence and stability analysis of spiky solutions for the Gierer-Meinhardt system with large reaction rates
We study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either stable or unstable, depending on the parameters. We fully classify the stability in this case. We characterise the destabilizing eigenfunctions in all cases. Numerical simulations are shown which are in full agreement with the analytical results
Existence and Stability of a Spike in the Central Component for a Consumer Chain Model
We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir
Suppression of low-energy Andreev states by a supercurrent in YBa_2Cu_3O_7-delta
We report a coherence-length scale phenomenon related to how the high-Tc
order parameter (OP) evolves under a directly-applied supercurrent. Scanning
tunneling spectroscopy was performed on current-carrying YBa_2Cu_3O_7-delta
thin-film strips at 4.2K. At current levels well below the theoretical
depairing limit, the low-energy Andreev states are suppressed by the
supercurrent, while the gap-like structures remain unchanged. We rule out the
likelihood of various extrinsic effects, and propose instead a model based on
phase fluctuations in the d-wave BTK formalism to explain the suppression. Our
results suggest that a supercurrent could weaken the local phase coherence
while preserving the pairing amplitude. Other possible scenarios which may
cause the observed phenomenon are also discussed.Comment: 6 pages, 4 figures, to appear in Physical Review
Modeling the IDV emissions of the BL Lac Objects with a Langevin type stochastic differential equation
In this paper, we introduce a simplified model for explaining the
observations of the optical intraday variability (IDV) of the BL Lac Objects.
We assume that the source of the IDV are the stochastic oscillations of an
accretion disk around a supermassive black hole. The Stochastic Fluctuations on
the vertical direction of the accretion disk are described by using a Langevin
type equation with a damping term and a random, white noise type force.
Furthermore, the preliminary numerical simulation results are presented, which
are based on the numerical analysis of the Langevin stochastic differential
equation.Comment: 4 pages, 4 figures, accepted for publication in J. Astrophys. Ast
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