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Full-Text Articles in Physical Sciences and Mathematics
Generalized Catalan Numbers From Hypergraphs, Paul E. Gunnells
Generalized Catalan Numbers From Hypergraphs, Paul E. Gunnells
Mathematics and Statistics Department Faculty Publication Series
The Catalan numbers Cn ∈ {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 Cn(m) , m >= 1, with m = 1 giving the …
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
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 …
Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed
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 …
Nonlinear Instabilities Of Multi-Site Breathers In Klein–Gordon Lattices, Jesús Cuevas Maraver, Panayotis G. Kevrekidis, Dmitry E. Pelinovsky
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 …
Rogue Waves In Nonlinear Schrodinger Models With Variable Coefficients : Application To Bose Einstein Condensates, J. S. He, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskasis
Rogue Waves In Nonlinear Schrodinger Models With Variable Coefficients : Application To Bose Einstein Condensates, J. S. He, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskasis
Mathematics and Statistics Department Faculty Publication Series
We explore the form of rogue waves solution sin a select set of case examples of non linear Schrodinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose Einstein condensates in different experimentally relevant settings. For these models, we identify exact rogue waves solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence of the rogue waves is identified. Additionally, the nontrivial fate of small numerically induced perturbations …
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Mathematics and Statistics Department Faculty Publication Series
In this work, we consider the dynamics of vector rogue waves and ark bright solitons in two component nonlinear Schrodinger equations with various physically motivated time dependent non linearity coefficients, as well as spatio temporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark bright boomeron like soliton solutions of the latter are converted back into ones of the original non autonomous model. Using direct numerical simulations we find that, in most cases, the rogue waves formation is rapidly followed by a modulational instability that leads …
Lattice Three Dimensional Skyrmions Revisited, E. G. Charalampidis, T. A. I, P. G. Kevrekidis
Lattice Three Dimensional Skyrmions Revisited, E. G. Charalampidis, T. A. I, P. G. Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, an 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 perturbation sis 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, …
Wormholes Threaded By Chiral Fields, Efstathios Charalampidis, Theodora Ioannidou, Burkhard Kleihaus, Jutta Kunz
Wormholes Threaded By Chiral Fields, Efstathios Charalampidis, Theodora Ioannidou, Burkhard Kleihaus, Jutta Kunz
Mathematics and Statistics Department Faculty Publication Series
We consider Lorentzian wormholes with a phantom field and chiral matter fields. The chiral fields are described by the non linear sigma model with or without a Skyrme term. When the gravitational coupling of the chiral fields is increased, the wormhole geometry changes. The single throat is replaced by a double throat with a belly inbetween. For a maximal value of the coupling, the radii of both throats reach zero. Then the interior part pinches off, leaving a closed universe and two (asympotically) flat spaces. A stability analysis shows that all wormholes threaded by chiral fields inherit the instability of …
Skyrmions, Rational Maps & Scaling Identities, E. G. Charalampidis, T. A. Ioannidou, N. S. Manton
Skyrmions, Rational Maps & Scaling Identities, E. G. Charalampidis, T. A. Ioannidou, N. S. Manton
Mathematics and Statistics Department Faculty Publication Series
Starting from approximate Skyrmion solutions obtained using the rational map ansatz, improved approximate Skyrmions are constructed using scaling arguments. Although the energy improvement is small, the change of shape clarifies whether the true Skyrmions are more oblate or prolate.