12 research outputs found
Collective oscillations in spatially modulated exciton-polariton condensate arrays
We study collective dynamics of interacting centers of exciton-polariton
condensation in presence of spatial inhomogeneity, as modeled by diatomic
active oscillator lattices. The mode formalism is developed and employed to
derive existence and stability criteria of plane wave solutions. It is
demonstrated that wave number mode with the binary elementary cell on a
diatomic lattice possesses superior existence and stability properties.
Decreasing net on-site losses (balance of dissipation and pumping) or
conservative nonlinearity favors multistability of modes, while increasing
frequency mismatch between adjacent oscillators detriments it. On the other
hand, spatial inhomogeneity may recover stability of modes at high
nonlinearities. Entering the region where all single-mode solutions are
unstable we discover subsequent transitions between localized quasiperiodic,
chaotic and global chaotic dynamics in the mode space, as nonlinearity
increases. Importantly, the last transition evokes the loss of synchronization.
These effects may determine lasing dynamics of interacting exciton-polariton
condensation centers.Comment: 9 pages, 3 figure
Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem
The Fermi-Pasta-Ulam problem was one of the first computational experiments.
It has stirred the physics community since, and resisted a simple solution for
half a century. The combination of straightforward simulations, efficient
computational schemes for finding periodic orbits, and analytical estimates
allows us to achieve significant progress. Recent results on -breathers,
which are time-periodic solutions that are localized in the space of normal
modes of a lattice and maximize the energy at a certain mode number, are
discussed, together with their relation to the Fermi-Pasta-Ulam problem. The
localization properties of a -breather are characterized by intensive
parameters, that is, energy densities and wave numbers. By using scaling
arguments, -breather solutions are constructed in systems of arbitrarily
large size. Frequency resonances in certain regions of wave number space lead
to the complete delocalization of -breathers. The relation of these features
to the Fermi-Pasta-Ulam problem are discussed.Comment: 19 pages, 9 figures, to appear in Am. J. Phy
q-Breathers and the Fermi-Pasta-Ulam Problem
The Fermi-Pasta-Ulam (FPU) paradox consists of the nonequipartition of energy
among normal modes of a weakly anharmonic atomic chain model. In the harmonic
limit each normal mode corresponds to a periodic orbit in phase space and is
characterized by its wave number . We continue normal modes from the
harmonic limit into the FPU parameter regime and obtain persistence of these
periodic orbits, termed here -Breathers (QB). They are characterized by time
periodicity, exponential localization in the -space of normal modes and
linear stability up to a size-dependent threshold amplitude. Trajectories
computed in the original FPU setting are perturbations around these exact QB
solutions. The QB concept is applicable to other nonlinear lattices as well.Comment: 4 pages, 4 figure
q-breathers in Discrete Nonlinear Schroedinger lattices
-breathers are exact time-periodic solutions of extended nonlinear systems
continued from the normal modes of the corresponding linearized system. They
are localized in the space of normal modes. The existence of these solutions in
a weakly anharmonic atomic chain explained essential features of the
Fermi-Pasta-Ulam (FPU) paradox. We study -breathers in one- two- and
three-dimensional discrete nonlinear Sch\"{o}dinger (DNLS) lattices --
theoretical playgrounds for light propagation in nonlinear optical waveguide
networks, and the dynamics of cold atoms in optical lattices. We prove the
existence of these solutions for weak nonlinearity. We find that the
localization of -breathers is controlled by a single parameter which depends
on the norm density, nonlinearity strength and seed wave vector. At a critical
value of that parameter -breathers delocalize via resonances, signaling a
breakdown of the normal mode picture and a transition into strong mode-mode
interaction regime. In particular this breakdown takes place at one of the
edges of the normal mode spectrum, and in a singular way also in the center of
that spectrum. A stability analysis of -breathers supplements these
findings. For three-dimensional lattices, we find -breather vortices, which
violate time reversal symmetry and generate a vortex ring flow of energy in
normal mode space.Comment: 19 pages, 9 figure
The Human Body as a Super Network: Digital Methods to Analyze the Propagation of Aging
Biological aging is a complex process involving multiple biological processes. These can be understood theoretically though considering them as individual networks—e.g., epigenetic networks, cell-cell networks (such as astroglial networks), and population genetics. Mathematical modeling allows the combination of such networks so that they may be studied in unison, to better understand how the so-called “seven pillars of aging” combine and to generate hypothesis for treating aging as a condition at relatively early biological ages. In this review, we consider how recent progression in mathematical modeling can be utilized to investigate aging, particularly in, but not exclusive to, the context of degenerative neuronal disease. We also consider how the latest techniques for generating biomarker models for disease prediction, such as longitudinal analysis and parenclitic analysis can be applied to as both biomarker platforms for aging, as well as to better understand the inescapable condition. This review is written by a highly diverse and multi-disciplinary team of scientists from across the globe and calls for greater collaboration between diverse fields of research
The Human Body as a Super Network: Digital Methods to Analyze the Propagation of Aging
Biological aging is a complex process involving multiple biological processes. These can be understood theoretically though considering them as individual networks—e.g., epigenetic networks, cell-cell networks (such as astroglial networks), and population genetics. Mathematical modeling allows the combination of such networks so that they may be studied in unison, to better understand how the so-called “seven pillars of aging” combine and to generate hypothesis for treating aging as a condition at relatively early biological ages. In this review, we consider how recent progression in mathematical modeling can be utilized to investigate aging, particularly in, but not exclusive to, the context of degenerative neuronal disease. We also consider how the latest techniques for generating biomarker models for disease prediction, such as longitudinal analysis and parenclitic analysis can be applied to as both biomarker platforms for aging, as well as to better understand the inescapable condition. This review is written by a highly diverse and multi-disciplinary team of scientists from across the globe and calls for greater collaboration between diverse fields of research
q-breathers in finite two- and three-dimensional nonlinear acoustic lattices
Nonlinear interaction between normal modes dramatically affects energy
equipartition, heat conduction and other fundamental processes in extended
systems. In their celebrated experiment Fermi, Pasta and Ulam (FPU, 1955)
observed that in simple one-dimensional nonlinear atomic chains the energy must
not always be equally shared among the modes. Recently, it was shown that exact
and stable time-periodic orbits, coined -breathers (QBs), localize the mode
energy in normal mode space in an exponential way, and account for many aspects
of the FPU problem. Here we take the problem into more physically important
cases of two- and three-dimensional acoustic lattices to find existence and
principally different features of QBs. By use of perturbation theory and
numerical calculations we obtain that the localization and stability of QBs is
enhanced with increasing system size in higher lattice dimensions opposite to
their one-dimensional analogues.Comment: 4 pages, 5 figure