Nonlinear Waves In Disordered Diatomic Granular Chains, 2010 University of Massachusetts - Amherst

#### 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 ...

Intrinsic Energy Localization Through Discrete Gap Breathers In One-Dimensional Diatomic Granular Crystals, 2010 University of Massachusetts - Amherst

#### 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 ...

Multibreather And Vortex Breather Stability In Klein–Gordon Lattices: Equivalence Between Two Different Approaches, 2010 University of Massachusetts - Amherst

#### 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 ...

Effects Of Long-Range Nonlinear Interactions In Double-Well Potentials, 2010 University of Massachusetts - Amherst

#### 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 ...

Discrete Breathers In One-Dimensional Diatomic Granular Crystals, 2010 University of Massachusetts - Amherst

#### 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 ...

Controlling The Transverse Instability Of Dark Solitons And Nucleation Of Vortices By A Potential Barrier, 2010 University of Massachusetts - Amherst

#### 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 ...

Collisional-Inhomogeneity-Induced Generation Of Matter-Wave Dark Solitons, 2010 University of Massachusetts - Amherst

#### 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 ...

Sequence And Structural Analysis Of The Chitinase Insertion Domain Reveals Two Conserved Motifs Involved In Chitin Binding, 2010 Old Dominion University

#### Sequence And Structural Analysis Of The Chitinase Insertion Domain Reveals Two Conserved Motifs Involved In Chitin Binding, Hai Li, Lesley H. Greene

*Chemistry & Biochemistry Faculty Publications*

Background: Chitinases are prevalent in life and are found in species including archaea, bacteria, fungi, plants, and animals. They break down chitin, which is the second most abundant carbohydrate in nature after cellulose. Hence, they are important for maintaining a balance between carbon and nitrogen trapped as insoluble chitin in biomass. Chitinases are classified into two families, 18 and 19 glycoside hydrolases. In addition to a catalytic domain, which is a triosephosphate isomerase barrel, many family 18 chitinases contain another module, i.e., chitinase insertion domain. While numerous studies focus on the biological role of the catalytic domain in chitinase ...

Stable Vortex–Bright-Soliton Structures In Two-Component Bose-Einstein Condensates, 2010 University of Massachusetts - Amherst

#### 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.

Stable Structures With High Topological Charge In Nonlinear Photonic Quasicrystals, 2010 University of Massachusetts - Amherst

#### 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.

Isothermic Submanifolds Of Symmetric $R$-Spaces, 2010 University of Massachusetts - Amherst

#### 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, 2010 University of Massachusetts - Amherst

#### 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.

A Note On The Non-Commutative Laplace–Varadhan Integral Lemma, 2010 University of Massachusetts - Amherst

#### 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.

Cells In Coxeter Groups I, 2010 University of Massachusetts - Amherst

#### Cells In Coxeter Groups I, M Belolipetsky, Pe Gunnells

*Mathematics and Statistics Department Faculty Publication Series*

No abstract provided.

Coarse-Graining Schemes For Stochastic Lattice Systems With Short And Long-Range Interactions, 2010 University of Massachusetts - Amherst

#### 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.

Torsion In The Cohomology Of Congruence Subgroups Of Sl(4, Z) And Galois Representations, 2010 University of Massachusetts - Amherst

#### 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.

Deterministic Equations For Stochastic Spatial Evolutionary Games, 2010 University of Massachusetts - Amherst

#### 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 ...

Metaplectic Ice, 2010 University of Massachusetts - Amherst

#### 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 ...

Automata And Cells In Affine Weyl Groups, 2010 University of Massachusetts - Amherst

#### 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.

Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, 2010 University of Massachusetts - Amherst

#### 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.