Aspects of Discrete Breathers and New Directions

We describe results concerning the existence proofs of Discrete Breathers (DBs) in the two classes of dynamical systems with optical linear phonons and with acoustic linear phonons. A standard approach is by continuation of DBs from an anticontinuous limit. A new approach, which is purely variational, is presented. We also review some numerical results on intraband DBs in random nonlinear systems. Some non-conventional physical applications of DBs are suggested. One of them is understanding slow relaxation properties of glassy materials. Another one concerns energy focusing and transport in biomolecules by targeted energy transfer of DBs. A similar theory could be used for describing targeted charge transfer of nonlinear electrons (polarons) and, more generally, for targeted transfer of several excitations (e.g. Davydov soliton).Comment: to appear in the Proceedings of NATO Advanced Research Workshop "Nonlinearity and Disorder: Theory and Applications", Tashkent,Uzbekistan,October 1-6, 200

Nonlinear Localization in Metamaterials

Metamaterials, i.e., artificially structured ("synthetic") media comprising weakly coupled discrete elements, exhibit extraordinary properties and they hold a great promise for novel applications including super-resolution imaging, cloaking, hyperlensing, and optical transformation. Nonlinearity adds a new degree of freedom for metamaterial design that allows for tuneability and multistability, properties that may offer altogether new functionalities and electromagnetic characteristics. The combination of discreteness and nonlinearity may lead to intrinsic localization of the type of discrete breather in metallic, SQUID-based, and ${\cal PT}-$symmetric metamaterials. We review recent results demonstrating the generic appearance of breather excitations in these systems resulting from power-balance between intrinsic losses and input power, either by proper initialization or by purely dynamical procedures. Breather properties peculiar to each particular system are identified and discussed. Recent progress in the fabrication of low-loss, active and superconducting metamaterials, makes the experimental observation of breathers in principle possible with the proposed dynamical procedures.Comment: 19 pages, 14 figures, Invited (Review) Chapte

Detrimental effects of natural vertical head gradients on chemical and water level measurements in observation wells: identification and control

It is well known that vertical head gradients exist in natural aquifer systems, and borehole flowmeter data have shown that such gradients commonly set up spontaneous vertical flows in monitoring wells, often called ambient flows. What has not been fully appreciated until recently is the serious detrimental effects such flows can have on solute concentration [Ground Water 39 (2001) 853] and hydraulic head measurements in monitoring wells. This communication explores the possibilities of diminishing ambient flows by increasing the hydraulic resistance to vertical flow within monitoring wells and limiting the penetration of such wells. Analyzed also are the surprising effects that vertical gradients may have on the equilibrium water level in a monitoring well. Results are based on collected data, numerical flow simulations, and hydraulic analysis in the near-well vicinity. Raising wellbore hydraulic resistance is of increasing importance and impact in thicker aquifers with higher horizontal hydraulic conductivities (K-h). A systematic analysis of screen penetration revealed that the reduction of ambient flow also depends on aquifer thickness. On a first order basis, the results for homogeneous aquifers may be used to estimate the behavior of a heterogeneous aquifer by computing a power-law average of the heterogeneous K-h(z). Finally, it is evident from the analysis of vertical gradients on well water levels that in the presence of sufficiently high gradients (partial derivativeh/partial derivativez > 0.5) it is physically possible for a well screen to be fully submerged below the water table, and yet have an internal water level below the top of the screen. Contrary to common perceptions, water levels in wells spanning the water table deviate significantly from the elevation of the formation water table when the local vertical gradient exceeds about 0.1. (C) 2003 Elsevier B.V. All rights reserved

Periodic Travelling Waves of Forced FPU Lattices

In this article, damped Fermi-Pasta-Ulam-type lattices driven by extended external forces are considered. The existence and uniqueness results of periodic travelling waves of the system are presented. The existence and the stability of periodic waves are also computed and discussed numerically. © 2013 Springer Science+Business Media New York

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as `generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding `sigmoid' curves similar to observables used in literature, e.g., the `spectral entropy' or various types of `correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the `time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the `time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as εc∼N^(−b), with b∈[1.5,2.5]. For `generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori