DNLS with Impurities

The past few years have witnessed an explosion of interest in discrete models and intrinsic localized modes (discrete breathers or solitons) that has been summarized in a number of recent reviews [1–3]. This growth has been motivated by numerous applications of nonlinear dynamical lattice models in areas as broad and diverse as the nonlinear optics of waveguide arrays [4], the dynamics of Bose–Einstein condensates in periodic potentials [5, 6], micro-mechanical models of cantilever arrays [7], or even simple models of the complex dynamics of the DNA double strand [8]. Arguably, the most prototypical model among the ones that emerge in these settings is the Discrete Nonlinear Schrödinger (DNLS) equation, the main topic of this book

Impulse-induced localized control of chaos in starlike networks

Locally decreasing the impulse transmitted by periodic pulses is shown to be a reliable method of taming chaos in starlike networks of dissipative nonlinear oscillators, leading to both synchronous periodic states and equilibria (oscillation death). Specifically, the paradigmatic model of damped kicked rotators is studied in which it is assumed that when the rotators are driven synchronously, i.e., all driving pulses transmit the same impulse, the networks display chaotic dynamics. It is found that the taming effect of decreasing the impulse transmitted by the pulses acting on particular nodes strongly depends on their number and degree of connectivity. A theoretical analysis is given explaining the basic physical mechanism as well as the main features of the chaos-control scenario.Ministerio de Economía y Competitividad (MINECO, Spain) Project No. FIS2012-34902 cofinanced by FEDER fundsJunta de Extremadura (JEx, Spain) Project No. GR1514

Supersonic Kinks in Coulomb lattices

There exist in nature examples of lattices of elements for which the interaction is repulsive, the elements are kept in place because different reasons, as border conditions, geometry (e.g., circular) and, certainly, the interaction with other elements in the system, which provides an external potential. A primer example are layered silicates as mica muscovite, where the potassium ions form a two dimensional lattice between silicate layers. We propose an extremely simplified model of this layer in order to isolate the properties of a repulsive lattice and study them. We find that they are extremely well suited for the propagation of supersonic kinks and multikinks. Theoretically, they may have as much energy and travel as fast as desired. This striking results suggest that the properties of repulsive lattices may be related with some yet not fully explained direct and indirect observations of lattice excitations in muscovite