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Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Discipline
- Keyword
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- Gibbs measures (2)
- Large deviation principle (2)
- Affine Grassmannian (1)
- Arrhenius dynamics (1)
- Birth-death process (1)
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- Bose--Einstein condensates (1)
- Coarse-grained stochastic processes (1)
- Detailed balance (1)
- Discrete nonlinear Schrödinger equation; Discrete solitons; Existence and stability; Lyapunov–Schmidt reductions (1)
- Domain wall; Soliton; Matter waves; Optical lattice; Bose–Einstein condensation (1)
- Equilibrium macrostates (1)
- Equivalence of ensembles (1)
- Fermi gas (1)
- First-order phase transition (1)
- Generalized canonical ensemble (1)
- Gibbs-KMS states (1)
- Hamiltonian PDE's (1)
- Hamiltonian systems (1)
- Heat conduction (1)
- High temperature (1)
- Kinetic Monte Carlo method (1)
- Klein–Gordon; Nonlinear Schrödinger; Discrete models; Long-range interactions; Peakons; Solitary waves; Stability analysis (1)
- Langlands dual group (1)
- Low regularity solutions (1)
- Microcanonical entropy (1)
- Monte Carlo simulations (1)
- Multiple scale perturbation theory (1)
- Nonequilibrium statistical mechanics (1)
- One dimension (1)
- Polarons; Discrete breathers; Polarobreathers (1)
Articles 1 - 30 of 35
Full-Text Articles in Physical Sciences and Mathematics
Complete Intersections In Toric Ideals, Eduardo Cattani, Raymond Curran, Alicia Dickenstein
Complete Intersections In Toric Ideals, Eduardo Cattani, Raymond Curran, Alicia Dickenstein
Mathematics and Statistics Department Faculty Publication Series
We present examples which show that in dimension higher than one or codimension higher than two, there exist toric ideals IA such that no binomial ideal contained in IA and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.
Domain Walls Of Single-Component Bose–Einstein Condensates In External Potentials, Pg Kevrekidis
Domain Walls Of Single-Component Bose–Einstein Condensates In External Potentials, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We demonstrate the possibility of creating domain walls described by a single component Gross–Pitaevskii equation with attractive interactions, in the presence of an optical–lattice potential. While it is found that the domain wall is unstable in an infinite system, we show that the external magnetic trap can stabilize it. Stable solutions also include “twisted” domain walls, as well as asymmetric solitons. The results apply as well to spatial solitons in planar nonlinear optical waveguides with transverse modulation of the refractive index.
Spontaneous Symmetry Breaking In Photonic Lattices: Theory And Experiment, Pg Kevrekidis
Spontaneous Symmetry Breaking In Photonic Lattices: Theory And Experiment, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We examine an example of spontaneous symmetry breaking in a double-well waveguide with a symmetric potential. The ground state of the system beyond a critical power becomes asymmetric. The effect is illustrated numerically, and quantitatively analyzed via a Galerkin truncation that clearly shows the bifurcation from a symmetric to an asymmetric steady state. This phenomenon is also demonstrated experimentally when a probe beam is launched appropriately into an optically induced photonic lattice in a photorefractive material.
Equivariant Homology And K-Theory Of Affine Grassmannians And Toda Lattices, R Bezrukavnikov, M Finkelberg, I Mirkovic
Equivariant Homology And K-Theory Of Affine Grassmannians And Toda Lattices, R Bezrukavnikov, M Finkelberg, I Mirkovic
Mathematics and Statistics Department Faculty Publication Series
For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its …
Large Deviations In Quantum Lattice Systems: One-Phase Region, M Lenci, L Rey-Bellet
Large Deviations In Quantum Lattice Systems: One-Phase Region, M Lenci, L Rey-Bellet
Mathematics and Statistics Department Faculty Publication Series
We give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.
Iwasawa Invariants Of Galois Deformations, T Weston
Iwasawa Invariants Of Galois Deformations, T Weston
Mathematics and Statistics Department Faculty Publication Series
Fix a residual ordinary representation :GF→GLn(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg–Vatsal and Emerton–Pollack–Weston, we show that the Iwasawa invariants of Selmer groups of deformations of depends only on and the ramification of the deformation.
Stability Of Discrete Solitons In Nonlinear Schrödinger Lattices, D E. Pelinovsky, Pg Kevrekidis
Stability Of Discrete Solitons In Nonlinear Schrödinger Lattices, D E. Pelinovsky, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrödinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of in-phase or anti-phase excited nodes, separated by an arbitrary sequence of empty nodes. By using stability analysis, we prove that the discrete solitons are all unstable near the anti-continuum limit, except for the solitons, which consist of alternating anti-phase excited nodes. We classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons.
Nonlinear Lattice Dynamics Of Bose–Einstein Condensates, Mason Porter, R Carretero-Gonzalez, Pg Kevrekidis
Nonlinear Lattice Dynamics Of Bose–Einstein Condensates, Mason Porter, R Carretero-Gonzalez, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
The Fermi–Pasta–Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in nonmetallic 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 …
Discrete Solitons And Vortices On Anisotropic Lattices, Pg Kevrekidis
Discrete Solitons And Vortices On Anisotropic Lattices, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We consider the effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation that predicts that broad quasicontinuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons (“vortex crosses”) feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called “super-symmetric” intersite-centered vortices (“vortex squares”), with …
Modules Over The Small Quantum Group And Semi-Infinite Flag Manifold, S Arkhipov, R Bezrukavnikov, A Braverman, D Gaitsgory, I Mirkovic
Modules Over The Small Quantum Group And Semi-Infinite Flag Manifold, S Arkhipov, R Bezrukavnikov, A Braverman, D Gaitsgory, I Mirkovic
Mathematics and Statistics Department Faculty Publication Series
We develop a theory of perverse sheaves on the semi-infinite flag manifold G((t))/N((t)) · T[[t]], and show that the subcategory of Iwahori-monodromy perverse sheaves is equivalent to the regular block of the category of representations of the small quantum group at an even root of unity.
Kida’S Formula And Congruences, R Pollack, T Weston
Kida’S Formula And Congruences, R Pollack, T Weston
Mathematics and Statistics Department Faculty Publication Series
No abstract provided.
Real K-Flats Tangent To Quadrants In R^N, Frank Sottile, Thorsten Theobald
Real K-Flats Tangent To Quadrants In R^N, Frank Sottile, Thorsten Theobald
Mathematics and Statistics Department Faculty Publication Series
Let dk,n and #k,n denote the dimension and the degree of the Grassmannian Gk,n, respectively. For each 1 ≤ k ≤ n−2 there are 2dk,n ·#k,n (a priori complex) k-planes in Pn tangent to dk,n general quadratic hypersurfaces in Pn. We show that this class of enumerative problems is fully real, i.e., for 1 ≤ k ≤ n − 2 there exists a configuration of dk,n real quadrics in (affine) real space Rn so that all the mutually tangent k-flats are real.
Complete Analysis Of Phase Transitions And Ensemble Equivalence For The Curie-Weiss-Potts Model, M Costeniuc, Rs Ellis, H Touchette
Complete Analysis Of Phase Transitions And Ensemble Equivalence For The Curie-Weiss-Potts Model, M Costeniuc, Rs Ellis, H Touchette
Mathematics and Statistics Department Faculty Publication Series
Using the theory of large deviations, we analyze the phase transition structure of the Curie–Weiss–Potts spin model, which is a mean-field approximation to the nearest-neighbor Potts model. It is equivalent to the Potts model on the complete graph on n vertices. The analysis is carried out both for the canonical ensemble and the microcanonical ensemble. Besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles, we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates, relating these to concavity and support properties of the microcanonical entropy. The Curie–Weiss–Potts model …
Low Regularity Solutions To A Gently Stochastic Nonlinear Wave Equation In Nonequilibrium Statistical Mechanics, L Rey-Bellet, Le Thomas
Low Regularity Solutions To A Gently Stochastic Nonlinear Wave Equation In Nonequilibrium Statistical Mechanics, L Rey-Bellet, Le Thomas
Mathematics and Statistics Department Faculty Publication Series
We consider a system of stochastic partial differential equations modeling heat conduction in a non-linear medium. We show global existence of solutions for the system in Sobolev spaces of low regularity, including spaces with norm beneath the energy norm. For the special case of thermal equilibrium, we also show the existence of an invariant measure (Gibbs state).
The Generalized Canonical Ensemble And Its Universal Equivalence With The Microcanonical Ensemble, M Costeniuc, Rs Ellis, H Touchette, B Turkington
The Generalized Canonical Ensemble And Its Universal Equivalence With The Microcanonical Ensemble, M Costeniuc, Rs Ellis, H Touchette, B Turkington
Mathematics and Statistics Department Faculty Publication Series
This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C2, then universal equivalence of ensembles holds with g taken from a class …
Analysis Of Phase Transitions In The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, Pt Otto, H Touchette
Analysis Of Phase Transitions In The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, Pt Otto, H Touchette
Mathematics and Statistics Department Faculty Publication Series
In this paper we give a complete analysis of the phase transitions in the mean-field Blume-Emery-Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of …
Nonequivalent Ensembles And Metastability, H Touchett, Rs Ellis
Nonequivalent Ensembles And Metastability, H Touchett, Rs Ellis
Mathematics and Statistics Department Faculty Publication Series
This paper reviews a number of fundamental connections that exist between nonequivalent microcanonical and canonical ensembles, the appearance of first-order phase transitions in the canonical ensemble, and thermodynamic metastable behavior.
Statics, Dynamics, And Manipulations Of Bright Matter-Wave Solitons In Optical Lattices, Pg Kevrekidis
Statics, Dynamics, And Manipulations Of Bright Matter-Wave Solitons In Optical Lattices, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
Motivated by the recent experimental achievements in the work with Bose-Einstein condensates (BECs), we consider bright matter-wave solitons, in the presence of a parabolic magnetic trap and a spatially periodic optical lattice (OL), in the attractive BEC. We examine pinned states of the soliton and their stability by means of perturbation theory. The analytical predictions are found to be in good agreement with numerical simulations. We then explore possibilities to use a time-modulated OL as a means of stopping and trapping a moving soliton, and of transferring an initially stationary soliton to a prescribed position by a moving OL. We …
Stabilizing The Discrete Vortex Of Topological Charge S=2, Pg Kevrekidis
Stabilizing The Discrete Vortex Of Topological Charge S=2, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We study the instability of the discrete vortex with topological charge S=2 in a prototypical lattice model and observe its mediation through the central lattice site. Motivated by this finding, we analyze the model with the central site being inert. We identify analytically and observe numerically the existence of a range of linearly stable discrete vortices with S=2 in the latter model. The range of stability is comparable to that of the recently observed experimentally S=1 discrete vortex, suggesting the potential for observation of such higher charge discrete vortices.
Polarobreathers In Soft Potentials, J Cuevas, Pg Kevrekidis
Polarobreathers In Soft Potentials, J Cuevas, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We consider polarons in models of coupled electronic and vibrational degrees of freedom, in the presence of a soft nonlinear inter-particle potential (Morse potential). In particular, we focus on a a bound state of a polaron with a breather, a so-called “polarobreather”. We analyze the existence of this branch based on frequency resonance conditions and illustrate its stability using Floquet spectrum techniques. Multi-site solutions of this type are also obtained both in the stationary case (two-site polarons) and in the breathing case (two-site polarobreathers). We also obtain a different branch of solutions, namely a polaronic nanopteron.
Oscillations Of Dark Solitons In Trapped Bose-Einstein Condensates, Dmitry Pelinovsky, Dj Frantzeskakis, Pg Kevrekidis
Oscillations Of Dark Solitons In Trapped Bose-Einstein Condensates, Dmitry Pelinovsky, Dj Frantzeskakis, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We consider a one-dimensional defocusing Gross–Pitaevskii equation with a parabolic potential. Dark solitons oscillate near the center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multi-scale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of oscillations for the dark soliton of arbitrary amplitude. The second-order approximation predicts the nonlinear growth rate of the oscillation amplitude, which results in decay of the dark soliton. The results are compared with the previous publications and numerical computations.
Discrete Klein–Gordon Models With Static Kinks Free Of The Peierls–Nabarro Potential, S V. Dmitriev, Pg Kevrekidis
Discrete Klein–Gordon Models With Static Kinks Free Of The Peierls–Nabarro Potential, S V. Dmitriev, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
For the nonlinear Klein–Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are, therefore, free of the Peierls–Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein–Gordon models where slow kinks practically do not experience the action of the Peierls–Nabarro potential. Such kinks are not trapped by the lattice and …
Willmore Tori In The 4-Sphere With Nontrivial Normal Bundle, K Leschke, F Pedit, U Pinkall
Willmore Tori In The 4-Sphere With Nontrivial Normal Bundle, K Leschke, F Pedit, U Pinkall
Mathematics and Statistics Department Faculty Publication Series
No abstract provided.
Three-Dimensional Nonlinear Lattices: From Oblique Vortices And Octupoles To Discrete Diamonds And Vortex Cubes, R Carretero-Gonzalez, Pg Kevrekidis
Three-Dimensional Nonlinear Lattices: From Oblique Vortices And Octupoles To Discrete Diamonds And Vortex Cubes, R Carretero-Gonzalez, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We construct a variety of novel localized topological structures in the 3D discrete nonlinear Schrödinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from multipole patterns and diagonal vortices to vortex “cubes” (stack of two quasiplanar vortices) and “diamonds” (formed by two orthogonal vortices).
Persistence And Stability Of Discrete Vortices In Nonlinear Schrödinger Lattices, D E. Pelinovsky, Pg Kevrekidis
Persistence And Stability Of Discrete Vortices In Nonlinear Schrödinger Lattices, D E. Pelinovsky, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We study discrete vortices in the two-dimensional nonlinear Schrödinger lattice with small coupling between lattice nodes. The discrete vortices in the anti-continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non-zero charge. Using the Lyapunov–Schmidt reductions, we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes. An example of a square discrete contour is considered that includes the vortex cell (also known as the off-site vortex). We classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti-continuum limit. We predict analytically and …
Nonlinear Lattices Generated From Harmonic Lattices With Geometric Constraints, S Takeno, S. V. Dmitriev, Pg Kevrekidis
Nonlinear Lattices Generated From Harmonic Lattices With Geometric Constraints, S Takeno, S. V. Dmitriev, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
Geometrical constraints imposed on higher-dimensional harmonic lattices generally lead to nonlinear dynamical lattice models. Helical lattices obtained by such a procedure are shown to be described by sine- plus linear-lattice equations. The interplay between sinusoidal and quadratic potential terms in such models is shown to yield localized nonlinear modes identified as intrinsic resonant modes.
Motion Of Discrete Solitons Assisted By Nonlinearity Management, J Cuevas, B A. Malomed, Pg Kevrekidis
Motion Of Discrete Solitons Assisted By Nonlinearity Management, J Cuevas, B A. Malomed, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We demonstrate that time-periodic modulation of the nonlinearity coefficient in the discrete nonlinear Schrödinger equation strongly facilitates creation of traveling solitons in the lattice. We predict this possibility in a semiqualitative form analytically, and test it in direct numerical simulations. Systematic computations reveal several generic dynamical regimes, depending on the amplitude and frequency of the time modulation, and on the initial thrust which sets the soliton in motion. These regimes include irregular motion of the soliton, regular motion of a decaying one, and regular motion of a stable soliton. The motion may occur in both the straight and reverse directions, …
Localization Of Nonlinear Excitations In Curved Waveguides, Yu B. Gaididei, P L. Christiansen, Pg Kevrekidis
Localization Of Nonlinear Excitations In Curved Waveguides, Yu B. Gaididei, P L. Christiansen, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
Motivated by the examples of a curved waveguide embedded in a photonic crystal and cold atoms moving in a waveguide created by a spatially inhomogeneous electromagnetic field, we examine the effects of geometry in a 'quantum channel' of parabolic form. Starting with the linear case we derive exact as well as approximate expressions for the eigenvalues and eigenfunctions of the linear problem. We then proceed to the nonlinear setting and its stationary states in a number of limiting cases that allow for analytical treatment. The results of our analysis are used as initial conditions in direct numerical simulations of the …
Landau-Zener Tunneling Of Bose-Einstein Condensates In An Optical Lattice, V V. Konotop, Pg Kevrekidis
Landau-Zener Tunneling Of Bose-Einstein Condensates In An Optical Lattice, V V. Konotop, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
A theory of the nonsymmetric Landau-Zener tunneling of Bose-Einstein condensates in deep optical lattices is presented. It is shown that periodic exchange of matter between the bands is described by a set of linearly coupled nonlinear Schrödinger equations. The key role of the modulational instability in rendering the interband transitions irreversible is highlighted.
Quiver Coefficients Are Schubert Structure Constants, Anders Skovsted Buch, Frank Sottile, Alexander Yong
Quiver Coefficients Are Schubert Structure Constants, Anders Skovsted Buch, Frank Sottile, Alexander Yong
Mathematics and Statistics Department Faculty Publication Series
Buch and Fulton established a formula for the cohomology class of a quiver variety [9], which Buch later extended to K-theory [5]. The K-theory formula expresses a quiver class as an integer linear combination of products of stable Grothendieck polynomials. Quiver coefficients are the coefficients of this linear