Physical Sciences and Mathematics Commons^™

Phase Diagram, Stability And Magnetic Properties Of Nonlinear Excitations In Spinor Bose–Einstein Condensates, G. C. Katsimiga, S. I. Mistakidis, P. Schmelcher, P. G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We present the phase diagram, the underlying stability and magnetic properties as well as the dynamics of nonlinear solitary wave excitations arising in the distinct phases of a harmonically confined spinor F = 1 Bose-Einstein condensate. Particularly, it is found that nonlinear excitations in the form of dark-dark-bright solitons exist in the antiferromagnetic and in the easy-axis phase of a spinor gas, being generally unstable in the former while possessing stability intervals in the latter phase. Dark-bright-bright solitons can be realized in the polar and the easy-plane phases as unstable and stable configurations respectively; the latter phase can also feature …

Generalized Catalan Numbers From Hypergraphs, Paul E. Gunnells

Mathematics and Statistics Department Faculty Publication Series

The Catalan numbers C_n ∈ {1,1,2,5,14,42,…} form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we defi ne a generalization of the Catalan numbers. In fact we defi ne an infinite collection of generalizations C_n^(m) , m >= 1, with m = 1 giving the …

Immune Classification Of Osteosarcoma, Trang Le, Sumeyye Su, Leili Shahriyari

Mathematics and Statistics Department Faculty Publication Series

Tumor immune microenvironment has been shown to be important in predicting the tumor progression and the outcome of treatments. This work aims to identify different immune patterns in osteosarcoma and their clinical characteristics. We use the latest and best performing deconvolution method, CIBERSORTx, to obtain the relative abundance of 22 immune cells. Then we cluster patients based on their estimated immune abundance and study the characteristics of these clusters, along with the relationship between immune infiltration and outcome of patients. We find that abundance of CD8 T cells, NK cells and M1 Macrophages have a positive association with prognosis, while …

Discrete Solitons And Vortices In Anisotropic Hexagonal And Honeycomb Lattices, Q E. Hoq, Panayotis G. Kevrekidis, A R. Bishop

Mathematics and Statistics Department Faculty Publication Series

In the present work, we consider the self-focusing discrete nonlinear Schrödinger equation on hexagonal and honeycomb lattice geometries. Our emphasis is on the study of the effects of anisotropy, motivated by the tunability afforded in recent optical and atomic physics experiments. We find that multi-soliton and discrete vortex states undergo destabilizing bifurcations as the relevant anisotropy control parameter is varied. We quantify these bifurcations by means of explicit analytical calculations of the solutions, as well as of their spectral linearization eigenvalues. Finally, we corroborate the relevant stability picture through direct numerical computations. In the latter, we observe the prototypical manifestation …

Scattering Of Waves By Impurities In Precompressed Granular Chains, Panos Kevrekidis, Alejandro Martinez, Hiromi Yasuda, Eunho Kim, Mason Porter, Jinkyu Yang

Mathematics and Statistics Department Faculty Publication Series

We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the re ection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber k → ± π), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance—enabling full transmission at a particular …

Energy Criterion For The Spectral Stability Of Discrete Breathers, Panos Kevrekidis, Jesus Cuevas-Maraver, Dmitry Pelinovsky

Mathematics and Statistics Department Faculty Publication Series

No abstract provided.

Dark-Bright Soliton Interactions Beyond The Integrable Limit, G. Katsimiga, J. Stockhofe, Panos Kevrekidis, P. Schmelcher

Mathematics and Statistics Department Faculty Publication Series

In this work we present a systematic theoretical analysis regarding dark-bright solitons and their interactions, motivated by recent advances in atomic two-component repulsively interacting Bose-Einstein condensates. In particular, we study analytically via a two-soliton ansatz adopted within a variational formulation the interaction between two dark-bright solitons in a homogeneous environment beyond the integrable regime, by considering general inter/intra-atomic interaction coefficients. We retrieve the possibility of a fixed point in the case where the bright solitons are out of phase. As the inter-component interaction is increased, we also identify an exponential instability of the two-soliton state, associated with a subcritical pitchfork …

A Pt-Symmetric Dual-Core System With The Sine-Gordon Nonlinearity And Derivative Coupling, Jesus Cuevas-Maraver, Boris Malomed, Panos Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

As an extension of the class of nonlinear PT -symmetric models, we propose a system of sine-Gordon equations, with the PT symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in coupled Frenkel-Kontorova (FK) chains, while the cross-derivative coupling, which was not considered before, is induced by three-particle interactions, provided that the particles in the parallel FK chains move in different directions. Nonlinear modes are then studied in this system. In particular, kink-kink (KK) and kink-antikink (KA) complexes are explored …

Performing Hong-Ou-Mandel-Type Numerical Experiments With Repulsive Condensates: The Case Of Dark And Dark-Bright Solitons, Panos Kevrekidis, Zhi-Yuan Sun, Peter Kruger

Mathematics and Statistics Department Faculty Publication Series

The Hong-Ou-Mandel experiment leads indistinguishable photons simultaneously reach-ing a 50:50 beam splitter to emerge on the same port through two-photon interference.Motivated by this phenomenon, we consider numerical experiments of the same flavor forclassical, wave objects in the setting of repulsive condensates. We examine dark solitonsinteracting with a repulsive barrier, a case in which we find no significant asymmetries inthe emerging waves after the collision, presumably due to their topological nature. We alsoconsider case examples of two-component systems, where the dark solitons trap a brightstructure in the second-component (dark-bright solitary waves). For these, pronouncedasymmetries upon collision are possible for the non-topological …

Vector Dark-Antidark Solitary Waves In Multi-Component Bose-Einstein Condensates, Panos Kevrekidis, I. Danaila, M. Khamehchi, V. Gokhroo, P. Engels

Mathematics and Statistics Department Faculty Publication Series

Multi-component Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures. In recent theoretical work, the notion of magnetic solitons has been introduced. Here we generalize this concept to vector dark-antidark solitary waves in multi-component Bose-Einstein condensates. We first provide concrete experimental evidence for such states in an atomic BEC and subsequently illustrate the broader concept of these states, which are based on the interplay between miscibility and inter-component repulsion. Armed with this more general conceptual framework, we expand the notion of such states to higher dimensions presenting the possibility of both vortex-antidark states and ring-antidark-ring (dark soliton) states. We perform …

Gemini: A Computationally-Efficient Search Engine For Large Gene Expression Datasets, Timothy Defreitas, Hachem Saddiki, Patrick Flaherty

Mathematics and Statistics Department Faculty Publication Series

Background

Low-cost DNA sequencing allows organizations to accumulate massive amounts of genomic data and use that data to answer a diverse range of research questions. Presently, users must search for relevant genomic data using a keyword, accession number of meta-data tag. However, in this search paradigm the form of the query – a text-based string – is mismatched with the form of the target – a genomic profile.

Results

To improve access to massive genomic data resources, we have developed a fast search engine, GEMINI, that uses a genomic profile as a query to search for similar genomic profiles. GEMINI …

Collapse For The Higher-Order Nonlinear Schrödinger Equation, V. Achilleos, S. Diamantidis, D. J. Frantzeskakis, T. P. Horikis, N. I. Karachalios, P. G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schrödinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are …

Multifrequency And Edge Breathers In The Discrete Sine-Gordon System Via Subharmonic Driving: Theory, Computation And Experiment, F. Palmero, J. Han, L. Q. English, T. J. Alexander, P. G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We consider a chain of torsionally-coupled, planar pendula shaken horizontally by an external sinusoidal driver. It has been known that in such a system, theoretically modeled by the discrete sine-Gordon equation, intrinsic localized modes, also known as discrete breathers, can exist. Recently, the existence of multifrequency breathers via subharmonic driving has been theoretically proposed and numerically illustrated by Xu et al. (2014) [21]. In this paper, we verify this prediction experimentally. Comparison of the experimental results to numerical simulations with realistic system parameters (including a Floquet stability analysis), and wherever possible to analytical results (e.g. for the subharmonic response …

Formation Of Rarefaction Waves In Origami-Based Metamaterials, H. Yasuda, C. Chong, E. G. Charalampidis, P. G. Kevrekidis, J. Yang

Mathematics and Statistics Department Faculty Publication Series

We investigate the nonlinear wave dynamics of origami-based metamaterials composed of Tachi-Miura polyhedron (TMP) unit cells. These cells exhibit strain softening behavior under compression, which can be tuned by modifying their geometrical configurations or initial folded conditions. We assemble these TMP cells into a cluster of origami-based metamaterials, and we theoretically model and numerically analyze their wave transmission mechanism under external impact. Numerical simulations show that origami-based metamaterials can provide a prototypical platform for the formation of nonlinear coherent structures in the form of rarefaction waves, which feature a tensile wavefront upon the application of compression to the system. We …

Lattice Three-Dimensional Skyrmions Revisited, E G. Charalampidis, T A. Ioannidou, Panayotis G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, and a stationary point of a particular energy functional. This paper describes lattice analogues of the aforementioned skyrmions, namely a natural way of using the topological properties of the three dimensional continuum Skyrme model to achieve topological stability on the lattice. In particular, using fixed point iterations, numerically exact lattice skyrmions are constructed; and their stability under small perturbations is explored by means of linear stability analysis. While stable branches of such solutions are identified, it is also shown that they possess a particularly delicate bifurcation structure, …

Exciting And Harvesting Vibrational States In Harmonically Driven Granular Chains, Efstathios G. Charalampidis, Christopher Chong, Eunho Kim, Heetae Kim, F. Li, Panayotis G. Kevrekidis, J. Lydon, Chiara Daraio, Jianke Yang

Mathematics and Statistics Department Faculty Publication Series

This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multi-modal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the Nonlinear Schr¨odinger (NLS) equation predicts the corresponding modes fairly well. We propose that nonlinear multi-modal …

Nonlinear Waves In A Strongly Nonlinear Resonant Granular Chain, Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein

Mathematics and Statistics Department Faculty Publication Series

We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schrödinger (DpS) equation, which is established as controllably approximating the solutions …

Existence And Stability Of Pt-Symmetric Vortices In Nonlinear Two-Dimensional Square Lattices, Haitao Xu, P. G. Kevrekidis, Dmitry E. Pelinovsky

Mathematics and Statistics Department Faculty Publication Series

Vortices symmetric with respect to simultaneous parity and time reversing transformations are considered on the square lattice in the framework of the discrete nonlinear Schrödinger equation. The existence and stability of vortex configurations is analyzed in the limit of weak coupling between the lattice sites, when predictions on the elementary cell of a square lattice (i.e., a single square) can be extended to a large (yet finite) array of lattice cells. Our analytical predictions are found to be in good agreement with numerical computations.

Traveling Waves And Their Tails In Locally Resonant Granular Systems, H Xu, P G. Kevrekidis, A Stefanov

Mathematics and Statistics Department Faculty Publication Series

In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier transformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of …

Non-Holonomic Constraints And Their Impact On Discretizations Of Klein-Gordon Lattice Dynamical Models, Panayotis G. Kevrekidis, Vakhtang Putkaradze, Zoi Rapti

Mathematics and Statistics Department Faculty Publication Series

We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the “proper” continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known “standard” discretizations, even in the …

Stabilization Of Ring Dark Solitons In Bose-Einstein Condensates, Wenlong Wang, P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, Tasso J. Kaper, Manjun Ma

Mathematics and Statistics Department Faculty Publication Series

Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensates are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, …

Non-Conservative Variational Approximation For Nonlinear Schrödinger Equations., J. Rossi, R. Carretero-González, P. G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

Recently, Galley [Phys. Rev. Lett. 110, 174301 (2013)] proposed an initial value problem formulation of Hamilton’s principle applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrödinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems; namely, the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models …

Superdiffusive Transport And Energy Localization In Disordered Granular Crystals, Alejandro J. Martínez, Panayotis G. Kevrekidis, Mason A. Porter

Mathematics and Statistics Department Faculty Publication Series

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to be fundamentally different from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder: an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements), and for two families of initial conditions: displacement perturbations and velocity perturbations. We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics strongly depends on the initial condition. …

Kink Scattering From A Parity-Time-Symmetric Defect In The Φ 4 Model, Danial Saadatmand, Sergey V. Dmitriev, D. I. Borisov, Panayotis G. Kevrekidis, Minnekhan A. Fatykhov, Kurosh Javidan

Mathematics and Statistics Department Faculty Publication Series

In this paper, we study the ϕ⁴ kink scattering from a spatially localized PT-symmetric defect and the effect of the kink’s internal mode (IM) is discussed. It is demonstrated that if a kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. This asymmetry is a principal finding of the present work. Similar to the case of the sine-Gordon kink studied earlier, it is found that the ϕ⁴ kink approaching the defect from the gain side always passes through the …

Solitary Waves In A Discrete Nonlinear Dirac Equation, Jesús Cuevas–Maraver, Panayotis G. Kevrekidis, Avadh Saxena

Mathematics and Statistics Department Faculty Publication Series

In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross–Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two- and three-site excitations playing a role analogous to one- and two-site excitations, respectively, of the discrete nonlinear Schrödinger analogue of the model. Stability exchanges between the two- and three-site states …

Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed

Mathematics and Statistics Department Faculty Publication Series

We study a two component nonlinear Schrodinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes …

Effects Of Interactions On The Generalized Hong–Ou–Mandel Effect, B. Gertjerenken, P. G. Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We numerically investigate the influence of interactions on the generalized Hong–Ou–Mandel (HOM) effect for bosonic particles in a (quasi-)one-dimensional set-up with weak harmonic confinement and show results for the cases of N = 2, N = 3 and N = 4 bosons interacting with a beam splitter, whose role is played by a δ-barrier. In particular, we focus on the effect of attractive interactions and compare the results with the repulsive case, as well as with the analytically available results for the non-interacting case (that we use as a benchmark). We observe a fermionization effect both for growing repulsive and …

Nonlinear Instabilities Of Multi-Site Breathers In Klein–Gordon Lattices, Jesús Cuevas Maraver, Panayotis G. Kevrekidis, Dmitry E. Pelinovsky

Mathematics and Statistics Department Faculty Publication Series

We explore the possibility of multi-site breather states in a nonlinear Klein–Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stabl e. The mechanism for this nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein–Gordon lattice with a soft (Morse) and a …

Solitary Waves Of A Pt-Symmetric Nonlinear Dirac Equation, Jesús Cuevas–Maraver, Panayotis G. Kevrekidis, Avadh Saxena, Fred Cooper, Avinash Khare, Andrew Comech, Carl M. Bender

Mathematics and Statistics Department Faculty Publication Series

Abstract—In the present work, we consider a prototypical example of a PT -symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT -phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PT -symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the PT - transition threshold. In the latter, they degenerate into linear waves. We also examine the dynamics of …

Existence, Stability And Dynamics Of Discrete Solitary Waves In A Binary Waveguide Array, Y. Shen, Panayotis G. Kevrekidis, G. Srinivasan, A. B. Aceves

Mathematics and Statistics Department Faculty Publication Series

Recent work has explored binary waveguide arrays in the long-wavelength, near-continuum limit, here we examine the opposite limit, namely the vicinity of the so-called anti-continuum limit. We provide a systematic discussion of states involving one, two and three excited waveguides, and provide comparisons that illustrate how the stability of these states differ from the monoatomic limit of a single type of waveguide. We do so by developing a general theory which systematically tracks down the key eigenvalues of the linearized system. When we find the states to be unstable, we explore their dynamical evolution through direct numerical simulations. The latter …