Mathematics and Statistics Department Faculty Publication SeriesCopyright (c) 2016 University of Massachusetts - Amherst All rights reserved.
http://scholarworks.umass.edu/math_faculty_pubs
Recent documents in Mathematics and Statistics Department Faculty Publication Seriesen-usWed, 14 Sep 2016 02:03:50 PDT3600Exciting and Harvesting Vibrational States in Harmonically Driven Granular Chains
http://scholarworks.umass.edu/math_faculty_pubs/1269
http://scholarworks.umass.edu/math_faculty_pubs/1269Mon, 12 Sep 2016 13:50:45 PDT
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 systems can be useful in vibration energy harvesting and discuss a prototypical framework for its realization. The electromechanical model we derive predicts accurately the conversion from mechanical to electrical energy observed in the experiments.
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Charalampidis, Efstathios G. et al.Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals
http://scholarworks.umass.edu/math_faculty_pubs/1268
http://scholarworks.umass.edu/math_faculty_pubs/1268Mon, 12 Sep 2016 13:50:41 PDT
In this work, we consider time-periodic structures of trimer granular crystals consisting of alternate chrome steel and tungsten carbide spherical particles yielding a spatial periodicity of three. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system become chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.
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Charalampidis, E. G. et al.Dark bright solitons in coupled nonlinear Schrodinger equations with unequal dispersion coefficients
http://scholarworks.umass.edu/math_faculty_pubs/1267
http://scholarworks.umass.edu/math_faculty_pubs/1267Mon, 12 Sep 2016 13:50:36 PDT
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 of potential stability, not only of the single peak ground state (the dark bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identifies as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although out principal focus is n the homogeneous setting, we also briefly touch upon the counter intuitive impact of the potential presence of a parabolic trap on the states of interest.
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Charalampidis, E. G. et al.Lattice Three Dimensional Skyrmions Revisited
http://scholarworks.umass.edu/math_faculty_pubs/1266
http://scholarworks.umass.edu/math_faculty_pubs/1266Mon, 12 Sep 2016 13:50:32 PDT
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, especially so in the vicinity of the continuum limit. The corresponding bifurcation diagram is elucidated and a prescription for selecting the branch asymptoting to the well known continuum limit is given. Finally, the robustness of the spectrally stable solutions is corroborated by the virtue of direct numerical simulations.
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Charalampidis, E. G. et al.Vector rogue waves and dark bright boomeronic solitons in autonomous and non autonomous settings
http://scholarworks.umass.edu/math_faculty_pubs/1265
http://scholarworks.umass.edu/math_faculty_pubs/1265Mon, 12 Sep 2016 13:50:27 PDT
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 to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.
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Mareeswaran, R. Babu et al.Rogue Waves in Nonlinear Schrodinger Models with Variable Coefficients : Application to Bose Einstein Condensates
http://scholarworks.umass.edu/math_faculty_pubs/1264
http://scholarworks.umass.edu/math_faculty_pubs/1264Mon, 12 Sep 2016 13:50:22 PDT
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 to the exact rogue waves solutions is also discussed.
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He, J. S. et al.Wormholes Threaded by Chiral Fields
http://scholarworks.umass.edu/math_faculty_pubs/1263
http://scholarworks.umass.edu/math_faculty_pubs/1263Mon, 12 Sep 2016 13:50:18 PDT
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 the Ellis wormhole.
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Charalampidis, Efstathios et al.Skyrmions, Rational Maps & Scaling Identities
http://scholarworks.umass.edu/math_faculty_pubs/1262
http://scholarworks.umass.edu/math_faculty_pubs/1262Mon, 12 Sep 2016 13:50:13 PDT
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.
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Charalampidis, E. G. et al.Formation of rarefaction waves in origami based metamaterials
http://scholarworks.umass.edu/math_faculty_pubs/1261
http://scholarworks.umass.edu/math_faculty_pubs/1261Mon, 12 Sep 2016 13:44:39 PDT
We investigate the nonlinear wave dynamics of origami based metamaterials compose 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 dorm of rarefaction waves, which feature a tensile wavefront upon the application of compression to the system. We also demonstrate the existence of numerically exact traveling rarefaction waves. Origami based metamaterials can be highly useful for mitigating shock waves, potentially enabling a wide variety of engineering applications.
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Yasuda, H. et al.