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University of Massachusetts Amherst
Mathematics and Statistics Department Faculty Publication Series
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Full-Text Articles in Physical Sciences and Mathematics
An Introduction To The Thermodynamic And Macrostate Levels Of Nonequivalent Ensembles, H Touchette, Rs Ellis, B Turkington
An Introduction To The Thermodynamic And Macrostate Levels Of Nonequivalent Ensembles, H Touchette, Rs Ellis, B Turkington
Mathematics and Statistics Department Faculty Publication Series
This short paper presents a nontechnical introduction to the problem of nonequivalent microcanonical and canonical ensembles. Both the thermodynamic and the macrostate levels of definition of nonequivalent ensembles are introduced. The many relationships that exist between these two levels are also explained in simple physical terms.
Thermodynamic Versus Statistical Nonequivalence Of Ensembles For The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, H Touchette, B Turkington
Thermodynamic Versus Statistical Nonequivalence Of Ensembles For The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, H Touchette, B Turkington
Mathematics and Statistics Department Faculty Publication Series
We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical …
Rich Example Of Geometrically Induced Nonlinearity: From Rotobreathers And Kinks To Moving Localized Modes And Resonant Energy Transfer, Pg Kevrekidis, Et. Al
Rich Example Of Geometrically Induced Nonlinearity: From Rotobreathers And Kinks To Moving Localized Modes And Resonant Energy Transfer, Pg Kevrekidis, Et. Al
Mathematics and Statistics Department Faculty Publication Series
We present an experimentally realizable, simple mechanical system with linear interactions whose geometric nature leads to nontrivial, nonlinear dynamical equations. The equations of motion are derived and their ground state structures are analyzed. Selective “static” features of the model are examined in the context of nonlinear waves including rotobreathers and kinklike solitary waves. We also explore “dynamic” features of the model concerning the resonant transfer of energy and the role of moving intrinsic localized modes in the process.
Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations And Quasiperiodic Solutions To Oscillating Domain Walls And Spiral Waves, B Deconinck, Pg Kevrekidis
Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations And Quasiperiodic Solutions To Oscillating Domain Walls And Spiral Waves, B Deconinck, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In this paper, an exact unitary transformation is examined that allows for the construction of solutions of coupled nonlinear Schrödinger equations with additional linear field coupling, from solutions of the problem where this linear coupling is absent. The most general case where the transformation is applicable is identified. We then focus on the most important special case, namely the well-known Manakov system, which is known to be relevant for applications in Bose-Einstein condensates consisting of different hyperfine states of 87Rb. In essence, the transformation constitutes a distributed, nonlinear as well as multi-component generalization of the Rabi oscillations between two-level atomic …
Dynamics Of Shallow Dark Solitons In A Trapped Gas Of Impenetrable Bosons, D J. Frantzeskakis, N P. Proukakis, Pg Kevrekidis
Dynamics Of Shallow Dark Solitons In A Trapped Gas Of Impenetrable Bosons, D J. Frantzeskakis, N P. Proukakis, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
The dynamics of linear and nonlinear excitations in a Bose gas in the Tonks-Girardeau regime with longitudinal confinement are studied within a mean-field theory of quintic nonlinearity. A reductive perturbation method is used to demonstrate that the dynamics of shallow dark solitons, in the presence of an external potential, can effectively be described by a variable-coefficient Korteweg–de Vries equation. The soliton oscillation frequency is analytically obtained to be equal to the axial trap frequency, in agreement with numerical predictions obtained by Busch et al. [J. Phys. B 36, 2553 (2003)] via the Bose-Fermi mapping. We obtain analytical expressions for the …
Some Results About Geometric Whittaker Model, R Bezrukavnikov, A Braverman, I Mirkovic
Some Results About Geometric Whittaker Model, R Bezrukavnikov, A Braverman, I Mirkovic
Mathematics and Statistics Department Faculty Publication Series
Let G be an algebraic reductive group over a field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of ℓ-adic sheaves on X with respect to a generic character commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary G-variety and we prove the above property for all -equivariant sheaves on X where is the unipotent radical of an …
Breather Statics And Dynamics In Klein-Gordon Chains With A Bend, J Cuevas, Pg Kevrekidis
Breather Statics And Dynamics In Klein-Gordon Chains With A Bend, J Cuevas, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also examine dynamical effects including the scattering of mobile localized modes (discrete breathers) off of such a geometric structure. The potential outcomes of such numerical experiments (including transmission, trapping within the bend as well as reflection) are highlighted and qualitatively explained. …
Character Sheaves On Reductive Lie Algebras, I Mirkovic
Character Sheaves On Reductive Lie Algebras, I Mirkovic
Mathematics and Statistics Department Faculty Publication Series
The paper develops a linearized notion of Lusztig's character sheaves (on Lie algebras rather then on groups), which contains Lusztig's class of character sheaves on Lie algebras. The theory is independent of the characteristic p of the field, and we use it to provide elementary proofs of some results of Lusztig (for instance, the observation that on groups all cuspidal sheaves are character sheaves).
Avoiding Infrared Catastrophes In Trapped Bose-Einstein Condensates, Pg Kevrekidis, G Theocharis, Dj Frantzeskakis, A Trombettoni
Avoiding Infrared Catastrophes In Trapped Bose-Einstein Condensates, Pg Kevrekidis, G Theocharis, Dj Frantzeskakis, A Trombettoni
Mathematics and Statistics Department Faculty Publication Series
This paper is concerned with the long-wavelength instabilities (infrared catastrophes) occurring in Bose-Einstein condensates (BECs). We examine the modulational instability in “cigar-shaped” (one-dimensional) attractive BECs and the transverse instability of dark solitons in “pancake” (two-dimensional) repulsive BECs. We suggest mechanisms, and give explicit estimates, on how to engineer the trapping conditions of the condensate to avoid such instabilities: the main result being that a tight enough trapping potential suppresses the instabilities present in the homogeneous limit. We compare the obtained estimates with numerical results and we highlight the relevant regimes of dynamical behavior.
Three-Dimensional Solitary Waves And Vortices In A Discrete Nonlinear Schrödinger Lattice, Panos Kevrekidis
Three-Dimensional Solitary Waves And Vortices In A Discrete Nonlinear Schrödinger Lattice, Panos Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger equation, we find discrete vortex solitons with various values of the topological charge S. Stability regions for the vortices with S=0,1,3 are investigated. The S=2 vortex is unstable and may spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of …
Variational Approach To The Modulational Instability, Z Rapti, Pg Kevrekidis
Variational Approach To The Modulational Instability, Z Rapti, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We study the modulational stability of the nonlinear Schrödinger equation using a time-dependent variational approach. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulational perturbations. Analyzing the ensuing ODE’s, we rederive the classical modulational instability criterion. The case (relevant to applications in optics and Bose-Einstein condensation) where the coefficients of the equation are time dependent, is also examined.
Static And Rotating Domain-Wall Cross Patterns In Bose-Einstein Condensates, B Malomed, H Nistazakis, D Frantzekakis, Pg Kevrekidis
Static And Rotating Domain-Wall Cross Patterns In Bose-Einstein Condensates, B Malomed, H Nistazakis, D Frantzekakis, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
For a Bose-Einstein condensate (BEC) in a two-dimensional (2D) trap, we introduce cross patterns, which are generated by the intersection of two domain walls (DWs) separating immiscible species, with opposite signs of the wave functions in each pair of sectors filled by the same species. The cross pattern remains stable up to the zero value of the immiscibility parameter ∣Δ∣, while simpler rectilinear (quasi-1D) DWs exist only for values of ∣Δ∣ essentially exceeding those in BEC mixtures (two spin states of the same isotope) currently available to the experiment. Both symmetric and asymmetric cross configurations are investigated, with equal or …
Solitary Wave Interactions In Dispersive Equations Using Manton’S Approach, Pg Kevrekidis
Solitary Wave Interactions In Dispersive Equations Using Manton’S Approach, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We generalize the approach first proposed by Manton [Nucl. Phys. B 150, 397 (1979)] to compute solitary wave interactions in translationally invariant, dispersive equations that support such localized solutions. The approach is illustrated using as examples solitons in the Korteweg–de Vries equation, standing waves in the nonlinear Schrödinger equation, and kinks as well as breathers of the sine-Gordon equation.
Resonant And Non-Resonant Modulated Amplitude Waves For Binary Bose–Einstein Condensates In Optical Lattices, Mason Porter, Pg Kevrekidis
Resonant And Non-Resonant Modulated Amplitude Waves For Binary Bose–Einstein Condensates In Optical Lattices, Mason Porter, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
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, …
Pattern Forming Dynamical Instabilities Of Bose–Einstein Condensates, Pg Kevrekidis
Pattern Forming Dynamical Instabilities Of Bose–Einstein Condensates, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in …
On Some Classes Of Mkdv Periodic Solutions, Pg Kevrekidis
On Some Classes Of Mkdv Periodic Solutions, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We obtain exact periodic solutions of the positive and negative modified Kortweg–de Vries (mKdV) equations. We examine the dynamical stability of these solitary wave lattices through direct numerical simulations. While the positive mKdV breather lattice solutions are found to be unstable, the two-soliton lattice solution of the same equation is found to be stable. Similarly, a negative mKdV lattice solution is found to be stable. We also touch upon the implications of these results for the KdV equation.
On The Modulational Instability Of The Nonlinear Schrödinger Equation With Dissipation, Z Rapti, Pg Kevrekidis
On The Modulational Instability Of The Nonlinear Schrödinger Equation With Dissipation, Z Rapti, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
The modulational instability (MI) of spatially uniform states in the nonlinear Schrödinger (NLS) equation is examined in the presence of higher-order dissipation. The study is motivated by results on the effects of three-body recombination in Bose-Einstein condensates (BECs), as well as by the important recent work of Segur et al. on the effects of linear damping in NLS settings. We show how the presence of even the weakest possible dissipation suppresses the instability on a longer time scale. However, on a shorter scale, the instability growth may take place, and a corresponding generalization of the MI criterion is developed. The …
Modulational Instabilities And Domain Walls In Coupled Discrete Nonlinear Schrödinger Equations, Z Rapti, A Trombettoni, Pg Kevrekidis
Modulational Instabilities And Domain Walls In Coupled Discrete Nonlinear Schrödinger Equations, Z Rapti, A Trombettoni, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We consider a system of two discrete nonlinear Schrödinger equations, coupled by nonlinear and linear terms. For various physically relevant cases, we derive a modulational instability criterion for plane-wave solutions. We also find and examine domain-wall solutions in the model with the linear coupling.
Parametric And Modulational Instabilities Of The Discrete Nonlinear Schrödinger Equation, Z Rapti, Pg Kevrekidis
Parametric And Modulational Instabilities Of The Discrete Nonlinear Schrödinger Equation, Z Rapti, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We examine the parametric and modulational instabilities arising in a non-autonomous, discrete nonlinear Schrödinger equation. The principal motivation for our study stems from the dynamics of Bose–Einstein condensates trapped in a deep optical lattice. We find that under periodic variations of the heights of the interwell barriers (or equivalently of the scattering length), in addition to the modulational instability, a window of parametric instability becomes available to the system. We explore this instability through multiple-scale analysis and identify it numerically. Its principal dynamical characteristic is that, typically, it develops over much larger times than the modulational instability, a feature that …
Tilting Exercises, A Beilinson, R Bezrukavnikov, I Mirkovic
Tilting Exercises, A Beilinson, R Bezrukavnikov, I Mirkovic
Mathematics and Statistics Department Faculty Publication Series
We discuss tilting objects in categories of perverse sheaves smooth along some stratification. In case of the Schubert stratification we show that the Radon transform interchanges tilting, projective, and injective objects, and prove Kapranov's conjecture on the Serre functor.