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Mason A. Porter

Publications and source records attributed to Mason A. Porter.

At least 235 records · Page 13Linked to original sources
arXiv2006

Nonlinearity Management in Optics: Experiment, Theory, and Simulation

We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air. By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented. We corroborate these experimental results with numerical simulations of the (2 + 1)-dimensional focusing cubic nonlinear Schrödinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.

nlin.PS↗
arXiv2005

Dynamics and Manipulation of Matter-Wave Solitons in Optical Superlattices

We analyze the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) ``dynamical superlattices'' can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic ``path-following'', in which a matter wave follows the time-dependent lattice substructure into oscillatory motion.

cond-mat.other↗
arXiv2005

One-Particle and Few-Particle Billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.

nlin.CD↗
arXiv2005

Bose-Einstein Condensates in Superlattices

We consider the Gross-Pitaevskii (GP) equation in the presence of periodic and quasiperiodic superlattices to study cigar-shaped Bose-Einstein condensates (BECs) in such potentials. We examine spatially extended wavefunctions in the form of modulated amplitude waves (MAWs). With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice wavenumber as well as ultrasubharmonic resonances that result from interactions of both wavenumbers of the superlattice. In each case, we determine the resulting system's equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions by direct simulations of the GP equation, identifying them as typically stable solutions of the model. We then study subharmonic resonances using Hamiltonian perturbation theory, tracing robust, spatio-temporally periodic patterns.

nlin.PS↗
arXiv2005

A network analysis of committees in the United States House of Representatives

Network theory provides a powerful tool for the representation and analysis of complex systems of interacting agents. Here we investigate the United States House of Representatives network of committees and subcommittees, with committees connected according to ``interlocks'' or common membership. Analysis of this network reveals clearly the strong links between different committees, as well as the intrinsic hierarchical structure within the House as a whole. We show that network theory, combined with the analysis of roll call votes using singular value decomposition, successfully uncovers political and organizational correlations between committees in the House without the need to incorporate other political information.

nlin.AO↗
arXiv2004

Nonlinear Lattice Dynamics of Bose-Einstein Condensates

The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic solids and develop ``experimental'' techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems--including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular, due to their widely ranging and easily manipulated dynamical apparatuses--with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials--provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.

nlin.PS↗
arXiv2004

Mathematical Models of Bipolar Disorder

We use limit cycle oscillators to model Bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about one percent of the United States adult population. We consider two nonlinear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here.

nlin.CD↗
arXiv2004

Resonant and Non-Resonant Modulated Amplitude Waves for Binary Bose-Einstein Condensates in Optical Lattices

We consider a system of two Gross-Pitaevskii (GP) equations, in the presence of an optical-lattice (OL) potential, coupled by both nonlinear and linear terms. This system describes a Bose-Einstein condensate (BEC) composed of two different spin states of the same atomic species, which interact linearly through a resonant electromagnetic field. In the absence of the OL, we find plane-wave solutions and examine their stability. In the presence of the OL, we derive a system of amplitude equations for spatially modulated states which are coupled to the periodic potential through the lowest-order subharmonic resonance. We determine this averaged system's equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions with direct simulations of the coupled GP equations. We find that symmetric (equal-amplitude) and asymmetric (unequal-amplitude) dual-mode resonant states are, respectively, stable and unstable. The unstable states generate periodic oscillations between the two condensate components, which is possible only because of the linear coupling between them. We also find four-mode states, but they are always unstable. Finally, we briefly consider ternary (three-component) condensates.

nlin.CD↗
arXiv2004

A Graphical User Interface to Simulate Classical Billiard Systems

Classical billiards constitute an important class of dynamical systems. They have not only been in used in mathematical disciplines such as ergodic theory, but their properties demonstrate fundamental physical phenomena that can be observed in laboratory settings. This document provides instructions for a Matlab module that simulates classical billiard systems. It is intended to be used as both a research and teaching tool. At present, the program efficiently simulates tables that are constructed entirely from line segments and elliptical arcs. It functions less reliably for tables with more complex boundary components. The program and documentation can be downloaded from \textit{http://www.math.gatech.edu/$\sim$mason/papers/}.

nlin.CD↗
arXiv2004

A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the intensity of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einstein condensates in detail using Hamiltonian perturbation theory and supporting numerical simulations.

nlin.CD↗
arXiv2004

The Bowl Championship Series: A Mathematical Review

We discuss individual components of the college football Bowl Championship Series, compare with a simple algorithm defined by random walks on a biased graph, attempt to predict whether the proposed changes will truly lead to increased BCS bowl access for non-BCS schools, and conclude by arguing that the true problem with the BCS Standings lies not in the computer algorithms, but rather in misguided addition.

physics.pop-ph↗
arXiv2004

Modulated Amplitude Waves in Bose-Einstein Condensates

We analyze spatio-temporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits (``modulated amplitude waves'') on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.

nlin.CD↗
arXiv2002

A Study on the Relevance of Information in Discriminative and Non-Discriminative Media

In this paper we compare the relevance of information obtained from "discriminative" media and from "non-discriminative" media. Discriminative media are the ones which accumulate and deliver information using a heuristic selection of it. This can be made by humans, or by artificial intelligent systems, exhibiting some form of "knowledge". Non-discriminative media just collect and return information without any distinction. This can also be made by humans or by artificial systems, but there is no "knowledge" involved in the process. We ranked the words occurring in an edited electronic publication specialized in complex systems research, and we found that they approximate a modified Zipf distribution. We compared occurrences of representative words from the distribution with the occurrences in non-discriminative media. We found that a non-discriminative medium (Google) has a higher variance from of our original distribution than a semi-discriminative one (NEC Research Index), even when both appear to have their own modified Zipf distribution. We conclude that discriminative media have a higher efficiency rating, at least in the area in which they specialize, than non-discriminative media. Using the same search method, the discriminative media should deliver more relevant information. This relevancy also depends on the skills of the user, but non-discriminative media are more sensitive to poor searching skills, as there is a higher probability of delivering irrelevant information. This leads us to suggest the incorporation of intelligent classifications in different media (such as the ones suggested by the Semantic Web project), in order to increase the relevance of the delivered information.

nlin.AO↗
arXiv2001

An Introduction to Quantum Chaos

Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.

nlin.CD↗
arXiv2001

Vibrating quantum billiards on Riemannian manifolds

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical ($\hbar \longrightarrow 0$) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.

nlin.CD↗
arXiv2001

The Radially Vibrating Spherical Quantum Billiard

We consider the radially vibrating spherical quantum billiard as a representative example of vibrating quantum billiards. We derive necessary conditions for quantum chaos in $d$-term superposition states. These conditions are symmetry relations corresponding to the relative quantum numbers of eigenstates considered pairwise. In this discussion, we give special attention to eigenstates with null angular momentum (for which the aforementioned condition is automatically satisfied). Moreover, when these necessary conditions are met, we observe numerically that there always exist parameter values that correspond to chaotic configurations. We focus our numerical studies on the ground and first excited states of the radially vibrating spherical billiard with null angular momentum eigenstates. We observe chaotic behavior in this configuration, which dispels the common belief that one is required to pass to the semiclassical ($\hbar \longrightarrow 0$) or high quantum number limits in order to meaningfully discuss quantum chaos. The results in the present paper are also of practical import, as the radially vibrating spherical quantum billiard may be used as a model for the quantum dot nanostructure, the Fermi accelerating sphere, and intra-nuclear particle behavior.

nlin.CD↗
arXiv2001

Bifurcations in one degree-of-vibration quantum billiards

We classify the local bifurcations of one dov quantum billiards, showing that only saddle-center bifurcations can occur. We analyze the resulting planar system when there is no coupling in the superposition state. In so doing, we also consider the global bifurcation structure. Using a double-well potential as a representative example, we demonstrate how to locate bifurcations in parameter space. We also discuss how to approximate the cuspidal loop using AUTO as well as how to cross it via continuation by detuning the dynamical system. Moreover, we show that when there is coupling, the resulting five-dimensional system--though chaotic--has a similar underlying structure. We verify numerically that both homoclinic orbits and cusps occur and provide an outline of an analytical argument for the existence of such homoclinic orbits. Small perturbations of the system reveal homoclinic tangles that typify chaotic behavior.

nlin.CD↗