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When Does Linear Stability Not Exclude Nonlinear Instability?, Panos Kevrekidis, D. E. Pelinovsky, A. Saxena

Panos Kevrekidis

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization’s internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic …

The Expected Total Curvature Of Random Polygons, Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, Clayton Shonkwiler

Robert Kusner

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.

We then consider the symmetric measure on closed polygons of fixed total length constructed by …

Stability And Tunneling Dynamics Of A Dark-Bright Soliton Pair In A Harmonic Trap, E. T. Karamatskos, J. Stockhofe, Panos Kevrekidis, P. Schmelcher

Panos Kevrekidis

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.91.043637

Mirror Symmetry For Log Calabi-Yau Surfaces I, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

We give a cononical sythetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y,D)

Highly Nonlinear Wave Propagation In Elastic Woodpile Periodic Structures, Panos Kevrekidis

Panos Kevrekidis

In the present work, we experimentally implement, numerically compute with, and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but is also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, …

Dark-Bright Solitons And Their Lattices In Atomic Bose-Einstein Condensates, D. Yan, F. Tsitoura, Panos Kevrekidis, D. J. Frantzeskakis

Panos Kevrekidis

In the present contribution, we explore a host of different stationary states, namely dark-bright solitons and their lattices, that arise in the context of multicomponent atomic Bose-Einstein condensates. The latter are modeled by systems of coupled Gross-Pitaevskii equations with general interaction (nonlinearity) coefficients gij. It is found that in some particular parameter ranges such solutions can be obtained in analytical form, however, numerically they are computed as existing in a far wider parametric range. Many features of the solutions under study, such as their analytical form without the trap or the stability and dynamical properties of one dark-bright soliton even …

Transitions From Order To Disorder In Multi-Dark And Multi-Dark-Bright Soliton Atomic Clouds, Wenlong Wang, Panos Kevrekidis

Panos Kevrekidis

We have performed a systematic study quantifying the variation of solitary wave behavior from that of an ordered cloud resembling a “crystalline” configuration to that of a disordered state that can be characterized as a soliton “gas.” As our illustrative examples, we use both one-component, as well as two-component, one-dimensional atomic gases very close to zero temperature, where in the presence of repulsive interatomic interactions and of a parabolic trap, a cloud of dark (dark-bright) solitons can form in the one- (two-) component system. We corroborate our findings through three distinct types of approaches, namely a Gross-Pitaevskii type of partial …

Pathwise Sensitivity Analysis In Transient Regimes, Georgios Arampatzis, Markos Katsoulakis, Yannis Pantazis

Markos Katsoulakis

The instantaneous relative entropy (IRE) and the corresponding instantaneous Fisher information matrix (IFIM) for transient stochastic processes are presented in this paper. These novel tools for sensitivity analysis of stochastic models serve as an extension of the well known relative entropy rate (RER) and the corresponding Fisher information matrix (FIM) that apply to stationary processes. Three cases are studied here, discrete-time Markov chains, continuous-time Markov chains and stochastic differential equations. A biological reaction network is presented as a demonstration numerical example.

Birational Geometry Of Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example [Spe13] of upper cluster algebras which are not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.

Generating Functions, Polynomials And Vortices With Alternating Signs In Bose-Einstein Condensates, Anna M. Barry, Farshid Hajir, P. G. Kevrekidis

Farshid Hajir

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of quasi-two-dimensional Bose–Einstein condensates in the large density (so-called Thomas–Fermi) limit, where the vortices can be treated as effective particles. In addition to the vortex–vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend …

Dynamics Of Vortex Dipoles In Anisotropic Bose-Einstein Condensates, Roy H. Goodman, Panos Kevrekidis, R. Carretero-Gonzalez

Panos Kevrekidis

We study the motion of a vortex dipole in a Bose--Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross--Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction …

Canonical Bases For Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel, Maxim Kontesevich

Paul Hacking

In GHK11, Conjecture 0.6, the first three authors conjectured the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, FG06, Conjecture 4.3. In particular, under …

Scattering Of Matter-Waves In Spatially Inhomogeneous Environments, F. Tsitoura, P. Kruger, Panos Kevrekidis, D. J. Frantzeskakis

Panos Kevrekidis

We study scattering of quasi-one-dimensional matter waves at an interface of two spatial domains, one with repulsive and one with attractive interatomic interactions. It is shown that the incidence of a Gaussian wave packet from the repulsive to the attractive region gives rise to generation of a soliton train. More specifically, the number of emergent solitons can be controlled, e.g., by the variation of the amplitude or the width of the incoming wave packet. Furthermore, we study the reflectivity of a soliton incident from the attractive region to the repulsive one. We find the reflection coefficient numerically and employ analytical …

Moduli Of Surfaces With An Anti-Canonical Cycle, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

We prove a global torelli theorem for pairs (Y,D) where Y is a smooth projective rational surface and D ∈ |−Ky | is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

Solitons And Vortices In Two-Dimensional Discrete Nonlinear Schrodinger Systems With Spatially Modulated Nonlinearity, Panos Kevrekidis

Panos Kevrekidis

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual “extended” unstaggered bright solitons, in which all sites are excited in the …