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Full-Text Articles in Physical Sciences and Mathematics
Maximally Inflected Real Rational Curves, Viatcheslav Kharlamov, Frank Sottile
Maximally Inflected Real Rational Curves, Viatcheslav Kharlamov, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in the Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.
Vortices In A Bose-Einstein Condensate Confined By An Optical Lattice, Pg Kevrekidis, R Carretero-Gonzalez, G Theocharis, Dj Frantzeskakis, Ba Malomed
Vortices In A Bose-Einstein Condensate Confined By An Optical Lattice, Pg Kevrekidis, R Carretero-Gonzalez, G Theocharis, Dj Frantzeskakis, Ba Malomed
Mathematics and Statistics Department Faculty Publication Series
We investigate the dynamics of vortices in repulsive Bose–Einstein condensates in the presence of an optical lattice (OL) and a parabolic magnetic trap. The dynamics is sensitive to the phase of the OL potential relative to the magnetic trap, and depends less on the OL strength. For the cosinusoidal OL potential, a local minimum is generated at the trap's centre, creating a stable equilibrium for the vortex, while in the case of the sinusoidal potential, the vortex is expelled from the centre, demonstrating spiral motion. Cases where the vortex is created far from the trap's centre are also studied, revealing …
Coupling Fields And Underlying Space Curvature: An Augmented Lagrangian Approach, Pg Kevrekidis, F L. Williams
Coupling Fields And Underlying Space Curvature: An Augmented Lagrangian Approach, Pg Kevrekidis, F L. Williams
Mathematics and Statistics Department Faculty Publication Series
We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space’s metric tensor to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which …
Domain Walls In Two-Component Dynamical Lattices, Pg Kevrekidis
Domain Walls In Two-Component Dynamical Lattices, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We introduce domain-wall (DW) states in the bimodal discrete nonlinear Schrödinger equation, in which the modes are coupled by cross-phase modulation (XPM). The results apply to an array of nonlinear optical waveguides carrying two different polarizations of light, or two different wavelengths, with anomalous intrinsic diffraction controlled by direction of the light beam, and to a string of drops of a binary Bose-Einstein condensate, trapped in an optical lattice. By means of continuation from various initial patterns taken in the anticontinuum (AC) limit, we find a number of different solutions of the DW type, for which different stability scenarios are …
Combinatorial Hopf Algebra And Generalized Dehn-Sommerville Relations, Frank Sottile
Combinatorial Hopf Algebra And Generalized Dehn-Sommerville Relations, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) : H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasi-symmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ) possesses two canonical Hopf subalgebras on which the character is even (respectively, …
Toric Ideals, Real Toric Varieties, And The Algebraic Moment Map, Frank Sottile
Toric Ideals, Real Toric Varieties, And The Algebraic Moment Map, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties. In particular, we explain the relation between linear precision and a particular linear projection we call the algebraic moment map.
On The Well-Posedness Of The Wave Map Problem In High Dimensions, Andrea Nahmod, Atanas Stefanov, Karen Uhlenbeck
On The Well-Posedness Of The Wave Map Problem In High Dimensions, Andrea Nahmod, Atanas Stefanov, Karen Uhlenbeck
Mathematics and Statistics Department Faculty Publication Series
We construct a gauge theoretic change of variables for the wave map from R × Rn into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - n ≥ 4 - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.
Averaging For Solitons With Nonlinearity Management, D E. Pelinovsky, Pg Kevrekidis
Averaging For Solitons With Nonlinearity Management, D E. Pelinovsky, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We develop an averaging method for solitons of the nonlinear Schrödinger equation with a periodically varying nonlinearity coefficient, which is used to effectively describe solitons in Bose-Einstein condensates, in the context of the recently proposed technique of Feshbach resonance management. Using the derived local averaged equation, we study matter-wave bright and dark solitons and demonstrate a very good agreement between solutions of the averaged and full equations.
On Quiver Varieties And Affine Grassmanians Of Type A, I Mirkovic, M Vybornov
On Quiver Varieties And Affine Grassmanians Of Type A, I Mirkovic, M Vybornov
Mathematics and Statistics Department Faculty Publication Series
We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. …
On Schrodinger Maps, Andrea R. Nahmod, Atanas Stefanov, Karen Uhlenbeck
On Schrodinger Maps, Andrea R. Nahmod, Atanas Stefanov, Karen Uhlenbeck
Mathematics and Statistics Department Faculty Publication Series
We study the question of well-posedness of the Cauchy problem for Schr¨odinger maps from R 1 ×R 2 to the sphere S 2 or to H2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schr¨odinger system of equations and then study this modified Schr¨odinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schr¨odinger map itself from it. In …
Stability Of Dark Solitons In A Bose-Einstein Condensate Trapped In An Optical Lattice, Pg Kevrekidis, R Carretero-Gonzalez, G Theocharis, Dj Frantzeskakis, Ba Malomed
Stability Of Dark Solitons In A Bose-Einstein Condensate Trapped In An Optical Lattice, Pg Kevrekidis, R Carretero-Gonzalez, G Theocharis, Dj Frantzeskakis, Ba Malomed
Mathematics and Statistics Department Faculty Publication Series
We investigate the stability of dark solitons (DSs) in an effectively one-dimensional Bose-Einstein condensate in the presence of the magnetic parabolic trap and an optical lattice (OL). The analysis is based on both the full Gross-Pitaevskii equation and its tight-binding approximation counterpart (discrete nonlinear Schrödinger equation). We find that DSs are subject to weak instabilities with an onset of instability mainly governed by the period and amplitude of the OL. The instability, if present, sets in at large times and it is characterized by quasiperiodic oscillations of the DS about the minimum of the parabolic trap.
Ring Dark Solitons And Vortex Necklaces In Bose-Einstein Condensates, G Theocharis, Dj Frantzeskakis, Pg Kevrekidis, Ba Malomed, Ys Kivshar
Ring Dark Solitons And Vortex Necklaces In Bose-Einstein Condensates, G Theocharis, Dj Frantzeskakis, Pg Kevrekidis, Ba Malomed, Ys Kivshar
Mathematics and Statistics Department Faculty Publication Series
We introduce the concept of ring dark solitons in Bose-Einstein condensates. We show that relatively shallow rings are not subject to the snake instability, but a deeper ring splits into a robust ringlike cluster of vortex pairs, which performs oscillations in the radial and azimuthal directions, following the dynamics of the original ring soliton.
Vortices In A Bose–Einstein Condensate Confined By An Optical Lattice, Pg Kevrekidis
Vortices In A Bose–Einstein Condensate Confined By An Optical Lattice, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We investigate the dynamics of vortices in repulsive Bose–Einstein condensates in the presence of an optical lattice (OL) and a parabolic magnetic trap. The dynamics is sensitive to the phase of the OL potential relative to the magnetic trap, and depends less on the OL strength. For the cosinusoidal OL potential, a local minimum is generated at the trap's centre, creating a stable equilibrium for the vortex, while in the case of the sinusoidal potential, the vortex is expelled from the centre, demonstrating spiral motion. Cases where the vortex is created far from the trap's centre are also studied, revealing …
Two-Soliton Collisions In A Near-Integrable Lattice System, S Dmitriev, Pg Kevrekidis
Two-Soliton Collisions In A Near-Integrable Lattice System, S Dmitriev, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We examine collisions between identical solitons in a weakly perturbed Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which corresponds to the Salerno model, and may be realized as an array of strongly overlapping nonlinear optical waveguides) or a quintic perturbation, or both. Complex dependences of the outcomes of the collisions on the initial phase difference between the solitons and location of the collision point are observed. Large changes of amplitudes and velocities of the colliding solitons are generated by weak perturbations, showing that the elasticity of soliton collisions in the AL model is fragile (for instance, the Salerno’s …
Instabilities And Bifurcations Of Nonlinear Impurity Modes, Pg Kevrekidis
Instabilities And Bifurcations Of Nonlinear Impurity Modes, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity, and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationary states near the impurity site, as well as examine both theoretical and numerical criteria for the transition between different localized states via a cascade of bifurcations.
Families Of Matter-Waves In Two-Component Bose-Einstein Condensates, Pg Kevrekidis
Families Of Matter-Waves In Two-Component Bose-Einstein Condensates, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We produce several families of solutions for two-component nonlinear Schrödinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs).
Toric Modular Forms Of Higher Weight, La Borisov, Pe Gunnells
Toric Modular Forms Of Higher Weight, La Borisov, Pe Gunnells
Mathematics and Statistics Department Faculty Publication Series
No abstract provided.
Transversals To Line Segments In Three-Dimensional Space, H Bronnimann, H Everett, S. Lazard, Frank Sottile, S. Whitesides
Transversals To Line Segments In Three-Dimensional Space, H Bronnimann, H Everett, S. Lazard, Frank Sottile, S. Whitesides
Mathematics and Statistics Department Faculty Publication Series
We completely describe the structure of the connected components of transversals to a collection of n line segments in R3. We show that n > 3 arbitrary line segments in R3 admit 0, 1, . . . , n or infinitely many line transversals. In the latter case, the transversals form up to n connected components.
Discrete Gap Solitons In A Diffraction-Managed Waveguide Array, Pg Kevrekidis
Discrete Gap Solitons In A Diffraction-Managed Waveguide Array, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
A model including two nonlinear chains with linear and nonlinear couplings between them, and opposite signs of the discrete diffraction inside the chains, is introduced. In the case of the cubic [ χ(3)] nonlinearity, the model finds two different interpretations in terms of optical waveguide arrays, based on the diffraction-management concept. A continuum limit of the model is tantamount to a dual-core nonlinear optical fiber with opposite signs of dispersions in the two cores. Simultaneously, the system is equivalent to a formal discretization of the standard model of nonlinear optical fibers equipped with the Bragg grating. A straightforward discrete second-harmonic-generation …
Dark-In-Bright Solitons In Bose-Einstein Condensates With Attractive Interactions, Pg Kevrekidis
Dark-In-Bright Solitons In Bose-Einstein Condensates With Attractive Interactions, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We demonstrate a possibility to generate localized states in effectively one-dimensional Bose-Einstein condensates with a negative scattering length in the form of a dark soliton in the presence of an optical lattice (OL) and/or a parabolic magnetic trap. We connect such structures with twisted localized modes (TLMs) that were previously found in the discrete nonlinear Schrödinger equation. Families of these structures are found as functions of the OL strength, tightness of the magnetic trap and chemical potential, and their stability regions are identified. Stable bound states of two TLMs are also found. In the case when the TLMs are unstable, …
Breathers On A Background: Periodic And Quasiperiodic Solutions Of Extended Discrete Nonlinear Wave Systems, Pg Kevrekidis
Breathers On A Background: Periodic And Quasiperiodic Solutions Of Extended Discrete Nonlinear Wave Systems, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In this paper we investigate the emergence of time-periodic and time quasiperiodic (sometimes infinitely long-lived and sometimes very long-lived or metastable) solutions of discrete nonlinear wave equations: discrete sine Gordon, discrete φ4 and discrete nonlinear Schrödinger equation (DNLS). The solutions we consider are periodic oscillations on a kink or standing wave breather background. The origin of these oscillations is the presence of internal modes, associated with the static ground state. Some of these modes are associated with the breaking of translational invariance, in going from a spatially continuous to a spatially discrete system. Others are associated with discrete modes which …
An Exploding Glass?, Pg Kevrekidis, Et. Al
An Exploding Glass?, Pg Kevrekidis, Et. Al
Mathematics and Statistics Department Faculty Publication Series
We propose a connection between self-similar, focusing dynamics in nonlinear partial differential equations (PDEs) and macroscopic dynamic features of the glass transition. In particular, we explore the divergence of the appropriate relaxation times in the case of hard spheres as the limit of random close packing is approached. We illustrate the analogy in the critical case, and suggest a “normal form” that can capture the onset of dynamic self-similarity in both phenomena.
Breather Lattice And Its Stabilization For The Modified Korteweg–De Vries Equation, Pg Kevrekidis
Breather Lattice And Its Stabilization For The Modified Korteweg–De Vries Equation, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We obtain an exact solution for the breather lattice solution of the modified Korteweg–de Vries equation. Numerical simulation of the breather lattice demonstrates its instability due to the breather-breather interaction. However, such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms.