Nonlinear Propagation of Light in One Dimensional Periodic Structures

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived {\it gap soliton} states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure

Non-equilibrium fixed points of coupled Ising models

Driven-dissipative systems are expected to give rise to non-equilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their non-equilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely non-equilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase---reminiscent of a liquid-gas transition---and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) $\mathbb{Z}_2$ symmetry. They, however, coalesce at a multicritical point, giving rise to a non-equilibrium model of coupled Ising-like order parameters described by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. Using a dynamical renormalization-group approach, we show that a pair of non-equilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the non-equilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths. Finally, we argue that this non-equilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.Comment: 19+11 pages, 7+9 figure

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker- Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure

Transfer of Antibiotic Resistance in Enterococcus faecalis. Modeling and Computational Study

Bacteria of the genus Enterococcus, commonly found in the intestinal tract, are the main cause of antibiotic-resistant infections that are acquired in hospitals[1], [2]. Donor cells that contain plasmid pCF10 have the ability to resist to antibiotics and are capable of transferring this plasmid to recipient cells. This transfer occurs via a rapid horizontal inducible conjugation regulated by peptide-mediated cell-cell signaling molecules (quorum sensing), known as cCF10 and iCF10. This quorum sensing system functions by producing low levels of an inducing substance that accumulates in the environments until a threshold is reached, at which point there is a change in cellular behavior. Cells of this type can either exist in the free floating form or in biofilms, which are composed of cells attached on biotic and abiotic surfaces. Complexity of the biofilm structure hinders and affects the exposures of cells to antibiotics and hence reduces treatment efficacy. Successful models of this mechanism can lead to useful techniques/methods in controlling or interfering with the plasmid transfer. Several efforts to model this phenomenon have been initiated and developed by our group in recent years. Recently, the collaborative experimental group in University of Minnesota has discovered new mechanisms that are associated with the system. This discover invalidates previous assumptions and hence requires modifications on both reactions and modeling assumptions. Moreover, various variables in the system have shown stiff behaviors that are much more challenging to work with. Explicit SDE, used in previous system, can be no longer capable of obtaining accurate solutions. For these reasons, this thesis presents new updated strategies to capture the drug resistance transfer in both Planktonic and biofilm environments. Since the two systems are inherently different in structure and physics, usage of varied modeling formulations for each environment is inevitable. Deterministic models are very simple and can be used to acquire a rough prediction of Planktonic environment. However, their simplicity also limits their capability of capturing large complex systems such as biofilms and other highly heterogeneous systems. Unfortunately, stochastic models can also carry a huge burden on CPU time. Therefore, another part of this thesis is dedicated to illustrate techniques, which can be used to reduce stochastic simulation time without losing accuracy. Successfully solving these two major problems together can potentially serve as a tool to gain knowledge about the system and eventually develop methods to treat/control this phenomenon

Quantum Hall transitions: An exact theory based on conformal restriction

We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, based on the theory of conformal restriction. This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the stochastic geometry of fractal curves and other stochastic geometrical fractal objects in 2D space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model. We show that the disorder-averaged PCCs are characterized by classical probabilities for certain geometric objects in the plane (pictures), occurring with positive statistical weights, that satisfy the crucial restriction property with respect to changes in the shape of the sample with absorbing boundaries. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of conformal restriction measures to relate the disorder-averaged PCCs to correlation functions of primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in 2D. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.Comment: Published versio

The Kardar-Parisi-Zhang equation and universality class

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- a non-linear stochastic partial differential equation -- known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.Comment: 57 pages, survey article, 7 figures, addition physics ref. added and typo's correcte