3,298 research outputs found
Delays in Open String Field Theory
We study the dynamics of light-like tachyon condensation in a linear dilaton
background using level-truncated open string field theory. The equations of
motion are found to be delay differential equations. This observation allows us
to employ well-established mathematical methods that we briefly review. At
level zero, the equation of motion is of the so-called retarded type and a
solution can be found very efficiently, even in the far light-cone future. At
levels higher than zero however, the equations are not of the retarded type. We
show that this implies the existence of exponentially growing modes in the
non-perturbative vacuum, possibly rendering light-like rolling unstable.
However, a brute force calculation using exponential series suggests that for
the particular initial condition of the tachyon sitting in the false vacuum in
the infinite light-cone past, the rolling is unaffected by the unstable modes
and still converges to the non-perturbative vacuum, in agreement with the
solution of Hellerman and Schnabl. Finally, we show that the growing modes
introduce non-locality mixing present with future, and we are led to conjecture
that in the infinite level limit, the non-locality in a light-like linear
dilaton background is a discrete version of the smearing non-locality found in
covariant open string field theory in flat space.Comment: 48 pages, 14 figures. v2: References added; Section 4 augmented by a
discussion of the diffusion equation; discussion of growing modes in Section
4 slightly expande
Global exponential stability for coupled systems of neutral delay differential equations
In this paper, a novel class of neutral delay differential equations (NDDEs) is presented. By using the Razumikhin method and Kirchhoff's matrix tree theorem in graph theory, the global exponential stability for such NDDEs is investigated. By constructing an appropriate Lyapunov function, two different kinds of sufficient criteria which ensure the global exponential stability of NDDEs are derived in the form of Lyapunov functions and coefficients of NDDEs, respectively. A numerical example is provided to demonstrate the effectiveness of the theoretical results
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
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