Physical Sciences and Mathematics Commons^™

Effects Of Long-Range Nonlinear Interactions In Double-Well Potentials, C Wang, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We consider the interplay of linear double-well-potential (DWP) structures and nonlinear longrange interactions of different types, motivated by applications to nonlinear optics and matter waves. We find that, while the basic spontaneous-symmetry-breaking (SSB) bifurcation structure in the DWP persists in the presence of the long-range interactions, the critical points at which the SSB emerges are sensitive to the range of the nonlocal interaction. We quantify the dynamics by developing a few-mode approximation corresponding to the DWP structure, and analyze the resulting system of ordinary differential equations and its bifurcations in detail. We compare results of this analysis with those produced …

Nonlinear Waves In Disordered Diatomic Granular Chains, Laurent Ponson, Nicholas Boechler, Yi M. Lai, Mason A. Porter, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a “spin,” so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: in low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of …

A Note On The Non-Commutative Laplace–Varadhan Integral Lemma, W De Roeck, C Maes, K Netocny, L Rey-Bellet

Mathematics and Statistics Department Faculty Publication Series

We continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian H an arbitrary mean field term is added, a polynomial function of the arithmetic mean of some local observables X and Y that do not necessarily commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz [10], we prove in general that the free energy is given by a variational principle over the range of the operators X and Y. As in [10], the result is a non-commutative extension of the Laplace–Varadhan asymptotic formula.

Matter-Wave Dark Solitons: Stochastic Versus Analytical Results, Sp Cockburn, He Nistazakis, Tp Horikis, Pg Kevrekidis, Np Proukakis, Dj Frantzeskakis

Mathematics and Statistics Department Faculty Publication Series

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for …

Existence And Stability Of Multisite Breathers In Honeycomb And Hexagonal Lattices, V Koukouloyannis, Pg Kevrekidis, Kjh Law, I Kourakis, Dj Frantzeskakis

Mathematics and Statistics Department Faculty Publication Series

We study the existence and stability of multisite discrete breathers in two prototypical non-square Klein–Gordon lattices, namely a honeycomb and a hexagonal one. In the honeycomb case we consider six-site configurations and find that for soft potential and positive coupling the out-of-phase breather configuration and the charge-two vortex breather are linearly stable, while the in-phase and charge-one vortex states are unstable. In the hexagonal lattice, we first consider three-site configurations. In the case of soft potential and positive coupling, the in-phase configuration is unstable and the charge-one vortex is linearly stable. The out-of-phase configuration here is found to always be …

Constructing Weyl Group Multiple Dirichlet Series, G Chinta, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

Let be a reduced root system of rank . A Weyl group multiple Dirichlet series for is a Dirichlet series in complex variables , initially converging for sufficiently large, that has meromorphic continuation to and satisfies functional equations under the transformations of corresponding to the Weyl group of . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others.

In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in …

Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell

Mathematics and Statistics Department Faculty Publication Series

In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms.

Stability And Dynamics Of Matter-Wave Vortices In The Presence Of Collisional Inhomogeneities And Dissipative Perturbations, S Middelkamp, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

In this work, the spectral properties of a singly charged vortex in a Bose–Einstein condensate confined in a highly anisotropic (disc-shaped) harmonic trap are investigated. Special emphasis is placed on the analysis of the so-called anomalous (negative energy) mode of the Bogoliubov spectrum. We use analytical and numerical techniques to illustrate the connection of the anomalous mode to the precession dynamics of the vortex in the trap. Effects due to inhomogeneous interatomic interactions and dissipative perturbations motivated by finite-temperature considerations are explored. We find that both of these effects may give rise to oscillatory instabilities of the vortex, which are …

Intrinsic Energy Localization Through Discrete Gap Breathers In One-Dimensional Diatomic Granular Crystals, G Theocharis, N Boechler, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy …

Dark Solitons In Cigar-Shaped Bose-Einstein Condensates In Double-Well Potentials, S Middelkamp, G Theocharis, Pg Kevrekidis, Dj Frantzeskakis, P Schmelcher

Mathematics and Statistics Department Faculty Publication Series

We study the statics and dynamics of dark solitons in a cigar-shaped Bose-Einstein condensate confined in a double-well potential. Using a mean-field model with a noncubic nonlinearity, appropriate to describe the dimensionality crossover regime from one- to three-dimensional, we obtain branches of solutions in the form of single and multiple dark soliton states, and study their bifurcations and stability. It is demonstrated that there exist dark soliton states which do not have a linear counterpart and we highlight the role of anomalous modes in the excitation spectra. Particularly, we show that anomalous mode eigenfrequencies are closely connected to the characteristic …

Dynamics Of Dark–Bright Solitons In Cigar-Shaped Bose–Einstein Condensates, S Middelkamp, J J. Chang, C Hammer, R Carretero-Gonzalez, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We explore the stability and dynamics of dark–bright (DB) solitons in two-component elongated Bose–Einstein condensates by developing effective one-dimensional vector equations and solving the three-dimensional Gross–Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the DB soliton on the atom number of its components is found; importantly, the wave may become dynamically unstable even in the 1D regime. As the atom number in the dark-soliton-supporting component is further increased, spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom. Moreover, the interactions of two DB solitons are investigated with an emphasis on …

Controlling The Transverse Instability Of Dark Solitons And Nucleation Of Vortices By A Potential Barrier, Manjun M, R. Carretero-Gonzalez, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC’s) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, …

Discrete Breathers In One-Dimensional Diatomic Granular Crystals, N Boechler, G Theocharis, S Job, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with …

Cells In Coxeter Groups I, M Belolipetsky, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

No abstract provided.

Torsion In The Cohomology Of Congruence Subgroups Of Sl(4, Z) And Galois Representations, A Ash, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

We report on the computation of torsion in certain homology the-ories of congruence subgroups of SL(4, Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels 31.

Distribution Of Eigenfrequencies For Oscillations Of The Ground State In The Thomas-Fermi Limit, Pg Kevrekidis, De Pelinovsky

Mathematics and Statistics Department Faculty Publication Series

In this work, we present a systematic derivation of the distribution of eigenfrequencies for oscillations of the ground state of a repulsive Bose-Einstein condensate in the semi-classical (Thomas-Fermi) limit. Our calculations are performed in one, two, and three-dimensional settings. Connections with the earlier work of Stringari, with numerical computations, and with theoretical expectations for invariant frequencies based on symmetry principles are also given.

Skyrmions, Rational Maps & Scaling Identities, E. G. Charalampidis, T. A. Ioannidou, N. S. Manton

Mathematics and Statistics Department Faculty Publication Series

Starting from approximate Skyrmion solutions obtained using the rational map ansatz, improved approximate Skyrmions are constructed using scaling arguments. Although the energy improvement is small, the change of shape clarifies whether the true Skyrmions are more oblate or prolate.

Linear Koszul Duality, I Mirkovic, S Riche

Mathematics and Statistics Department Faculty Publication Series

In this paper we construct, for F₁ and F₂ subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of _m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.

Stable Structures With High Topological Charge In Nonlinear Photonic Quasicrystals, K Law, A Saxena, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

Stable vortices with topological charges of 3 and 4 are examined numerically and analytically in photonic quasicrystals created by interference of five as well as eight beams, for cubic as well as saturable nonlinearities. Direct numerical simulations corroborate the analytical and numerical linear stability analysis predictions for such experimentally realizable structures.

Stable Vortex–Bright-Soliton Structures In Two-Component Bose-Einstein Condensates, K Law, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We report the numerical realization of robust two-component structures in 2D and 3D Bose-Einstein condensates with nontrivial topological charge in one component. We identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, due to the effective potential created by a stable vortex in the other component. The resulting vortex–bright-solitons, generalizations of the recently experimentally observed dark-bright solitons, are found to be very robust both in the homogeneous medium and in the presence of external confinement.

Multibreather And Vortex Breather Stability In Klein–Gordon Lattices: Equivalence Between Two Different Approaches, J Cuevas, V Koukouloyannis, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

In this work, we revisit the question of stability of multibreather configurations, i.e., discrete breathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. We present two methods that yield quantitative predictions about the Floquet multipliers of the linear stability analysis around such exponentially localized in space, time-periodic orbits, based on the Aubry band method and the MacKay effective Hamiltonian method and prove that their conclusions are equivalent. Subsequently, we showcase the usefulness of the methods by a series of case examples including one-dimensional multi-breathers, and two-dimensional vortex breathers in the case of a lattice of linearly coupled …

Multiple Atomic Dark Solitons In Cigar-Shaped Bose-Einstein Condensates, G Theocharis, A Weller, Jp Ronzheimer, C Gross, Mk Oberthaler, Pg Kevrekidis, Dj Frantzeskakis

Mathematics and Statistics Department Faculty Publication Series

We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system …

Short Pulse Equations And Localized Structures In Frequency Band Gaps Of Nonlinear Metamaterials, Nl Tsitsas, Tr Horikis, Y Shen, Panayotis Kevrekidis, N Whitaker, Dj Frantzeskakis

Mathematics and Statistics Department Faculty Publication Series

We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Two short-pulse equations (SPEs) are derived for the high- and low-frequency “band gaps” (where linear electromagnetic waves are evanescent) with linear effective permittivity <0 and permeability μ>0. The structure of the solutions of the SPEs is also briefly discussed, and connections with the soliton solutions of the nonlinear Schrödinger equation are made.

Collisional-Inhomogeneity-Induced Generation Of Matter-Wave Dark Solitons, C Wang, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We propose an experimentally relevant protocol for the controlled generation of matter-wave dark solitons in atomic Bose–Einstein condensates (BECs). In particular, using direct numerical simulations, we show that by switching-on a spatially inhomogeneous (step-like) change of the s-wave scattering length, it is possible to generate a controllable number of dark solitons in a quasi-one-dimensional BEC. A similar phenomenology is also found in the two-dimensional setting of “disk-shaped” BECs but, as the solitons are subject to the snaking instability, they decay into vortex structures. A detailed investigation of how the parameters involved affect the emergence and evolution of solitons and vortices …

Isothermic Submanifolds Of Symmetric $R$-Spaces, F Burstall, N Donaldson, F Pedit, U Pinkall

Mathematics and Statistics Department Faculty Publication Series

We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric $R$-spaces with essentially no loss of integrable structure.

Ruelle-Lanford Functions For Quantum Spin Systems, Y Ogata, L Rey-Bellet

Mathematics and Statistics Department Faculty Publication Series

We prove a large deviation principle for the expectation of macroscopic
observables in quantum (and classical) Gibbs states. Our proof is based
on Ruelle-Lanford functions [20, 34] and direct subadditivity arguments,
as in the classical case [23, 32], instead of relying on G¨artner-Ellis theorem,
and cluster expansion or transfer operators as done in the quantum case
in [21, 13, 27, 22, 16, 28]. In this approach we recover, expand, and unify
quantum (and classical) large deviation results for lattice Gibbs states. In
the companion paper [29] we discuss the characterization of rate functions
in terms of relative entropies.

Deterministic Equations For Stochastic Spatial Evolutionary Games, Sh Hwang, Ma Katsoulakis, L Rey-Bellet

Mathematics and Statistics Department Faculty Publication Series

In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error …

Metaplectic Ice, B Brubaker, D Bump, G Chinta, S Friedberg, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [5, 17], and we translate this description into an expression of the values of the Whittaker function as partition functions of a six-vertex model. Properties of theWhittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different …

Coarse-Graining Schemes For Stochastic Lattice Systems With Short And Long-Range Interactions, Ma Katsoulakis, P Plechac, L Rey-Bellet, D Tsagkarogiannis

Mathematics and Statistics Department Faculty Publication Series

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a d-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarse-grained lattice systems with short and long-range interactions.

Automata And Cells In Affine Weyl Groups, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

Let be an affine Weyl group, and let be a left, right, or two-sided Kazhdan-Lusztig cell in . Let be the set of all reduced expressions of elements of , regarded as a formal language in the sense of the theory of computation. We show that is a regular language. Hence, the reduced expressions of the elements in any Kazhdan-Lusztig cell can be enumerated by a finite state automaton.