250 research outputs found
Modulational Instability in Bose-Einstein Condensates under Feshbach Resonance Management
We investigate the modulational instability of nonlinear Schr{\"o}dinger
equations with periodic variation of their coefficients. In particular, we
focus on the case of the recently proposed, experimentally realizable protocol
of Feshbach Resonance Management for Bose-Einstein condensates. We derive the
corresponding linear stability equation analytically and we show that it can be
reduced to a Kronig-Penney model, which allows the determination of the windows
of instability. The results are tested numerically in the absence, as well as
in the presence of the magnetic trapping potential
On the Modulational Instability of the Nonlinear Schr\"odinger Equation with Dissipation
The modulational instability of spatially uniform states in the nonlinear
Schr\"odinger equation is examined in the presence of higher-order dissipation.
The study is motivated by results on the effects of three-body recombination in
Bose-Einstein condensates, as well as by the important recent work of Segur et
al. on the effects of linear damping in NLS settings. We show how the presence
of even the weakest possible dissipation suppresses the instability on a longer
time scale. However, on a shorter scale, the instability growth may take place,
and a corresponding generalization of the MI criterion is developed. The
analytical results are corroborated by numerical simulations. The method is
valid for any power-law dissipation form, including the constant dissipation as
a special case
Reaction-diffusion spatial modeling of COVID-19: Greece and Andalusia as case examples
We examine the spatial modeling of the outbreak of COVID-19 in two regions:
the autonomous community of Andalusia in Spain and the mainland of Greece. We
start with a 0D compartmental epidemiological model consisting of Susceptible,
Exposed, Asymptomatic, (symptomatically) Infected, Hospitalized, Recovered, and
deceased populations. We emphasize the importance of the viral latent period
and the key role of an asymptomatic population. We optimize model parameters
for both regions by comparing predictions to the cumulative number of infected
and total number of deaths via minimizing the norm of the difference
between predictions and observed data. We consider the sensitivity of model
predictions on reasonable variations of model parameters and initial
conditions, addressing issues of parameter identifiability. We model both
pre-quarantine and post-quarantine evolution of the epidemic by a
time-dependent change of the viral transmission rates that arises in response
to containment measures. Subsequently, a spatially distributed version of the
0D model in the form of reaction-diffusion equations is developed. We consider
that, after an initial localized seeding of the infection, its spread is
governed by the diffusion (and 0D model "reactions") of the asymptomatic and
symptomatically infected populations, which decrease with the imposed
restrictive measures. We inserted the maps of the two regions, and we imported
population-density data into COMSOL, which was subsequently used to solve
numerically the model PDEs. Upon discussing how to adapt the 0D model to this
spatial setting, we show that these models bear significant potential towards
capturing both the well-mixed, 0D description and the spatial expansion of the
pandemic in the two regions. Veins of potential refinement of the model
assumptions towards future work are also explored.Comment: 28 pages, 16 figures and 2 movie
Soliton Dynamics in Linearly Coupled Discrete Nonlinear Schr\"odinger Equations
We study soliton dynamics in a system of two linearly coupled discrete
nonlinear Schr\"odinger equations, which describe the dynamics of a
two-component Bose gas, coupled by an electromagnetic field, and confined in a
strong optical lattice. When the nonlinear coupling strengths are equal, we use
a unitary transformation to remove the linear coupling terms, and show that the
existing soliton solutions oscillate from one species to the other. When the
nonlinear coupling strengths are different, the soliton dynamics is numerically
investigated and the findings are compared to the results of an effective
two-mode model. The case of two linearly coupled Ablowitz-Ladik equations is
also investigated.Comment: to be published in Mathematics and Computers in Simulation,
proceedings of the fifth IMACS International Conference on Nonlinear
Evolution Equations and Wave Phenomena: Computation and Theory (Athens,
Georgia - April 2007
Non-Holonomic Constraints and Their Impact on Discretizations of Klein-Gordon Lattice Dynamical Models
We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the “proper” continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known “standard” discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important
Modulational and Parametric Instabilities of the Discrete Nonlinear Schr\"odinger Equation
We examine the modulational and parametric instabilities arising in a
non-autonomous, discrete nonlinear Schr{\"o}dinger equation setting. The
principal motivation for our study stems from the dynamics of Bose-Einstein
condensates trapped in a deep optical lattice. We find that under periodic
variations of the heights of the interwell barriers (or equivalently of the
scattering length), additionally to the modulational instability, a window of
parametric instability becomes available to the system. We explore this
instability through multiple-scale analysis and identify it numerically. Its
principal dynamical characteristic is that, typically, it develops over much
larger times than the modulational instability, a feature that is qualitatively
justified by comparison of the corresponding instability growth rates
Pattern Forming Dynamical Instabilities of Bose-Einstein Condensates: A Short Review
In this short topical review, we revisit a number of works on the
pattern-forming dynamical instabilities of Bose-Einstein condensates in one-
and two-dimensional settings. In particular, we illustrate the trapping
conditions that allow the reduction of the three-dimensional, mean field
description of the condensates (through the Gross-Pitaevskii equation) to such
lower dimensional settings, as well as to lattice settings. We then go on to
study the modulational instability in one dimension and the snaking/transverse
instability in two dimensions as typical examples of long-wavelength
perturbations that can destabilize the condensates and lead to the formation of
patterns of coherent structures in them. Trains of solitons in one-dimension
and vortex arrays in two-dimensions are prototypical examples of the resulting
nonlinear waveforms, upon which we briefly touch at the end of this review.Comment: 28 pages, 9 figures, publishe
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The role of mobility in the dynamics of the COVID-19 epidemic in Andalusia
Metapopulation models have been a popular tool for the study of epidemic spread over a network of highly populated nodes (cities, provinces, countries) and have been extensively used in the context of the ongoing COVID-19 pandemic. In the present work, we revisit such a model, bearing a particular case example in mind, namely that of the region of Andalusia in Spain during the period of the summer-fall of 2020 (i.e., between the first and second pandemic waves). Our aim is to consider the possibility of incorporation of mobility across the province nodes focusing on mobile-phone time dependent data, but also discussing the comparison for our case example with a gravity model, as well as with the dynamics in the absence of mobility. Our main finding is that mobility is key towards a quantitative understanding of the emergence of the second wave of the pandemic and that the most accurate way to capture it involves dynamic (rather than static) inclusion of time-dependent mobility matrices based on cell-phone data. Alternatives bearing no mobility are unable to capture the trends revealed by the data in the context of the metapopulation model considered herein
Lockdown Measures and their Impact on Single- and Two-age-structured Epidemic Model for the COVID-19 Outbreak in Mexico
The role of lockdown measures in mitigating COVID-19 in Mexico is investigated using a comprehensive
nonlinear ODE model. The model includes both asymptomatic and presymptomatic populations with the latter
leading to sickness (with recovery, hospitalization and death possibilities). We consider the situation involving
the imposed application of partial social distancing measures in the time series of interest and find optimal parametric fits to the time series of deaths (only), as well as to that of deaths and cumulative infections. We discuss
the merits and disadvantages of each approach, we interpret the parameters of the model and assess the realistic
nature of the parameters resulting from the optimization procedure. Importantly, we explore a model involving
two sub-populations (younger and older than a specific age), to more accurately reflect the observed impact as
concerns symptoms and behavior in different age groups. For definitiveness and to separate people that are (typically) in the active workforce, our partition of population is with respect to members younger vs. older than the
age of 65. The basic reproductive number of the model is computed for both the single- and the two-population
variant. Finally, we consider what would be the impact on the number of deaths and cumulative infections upon
imposition of partial lockdown (involving only the older population) and full lockdown (involving the entire
population)
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