Discrete breathers in Φ4 and related models

In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the 4 model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs. Our main emphasis is on the existence, and especially on the stability features of such solutions.We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this Chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored DB reviews) and associated future possibilities.AEI/FEDER, (UE) MAT2016- 79866-

Nonlinearity and Topology

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose-Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. Interaction among and collision of these nontrivial defects itself is a topic of great interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g. the Bogomolnyi decomposition. Some applications of this interplay, e.g. in nonreciprocal photonics as well as topological materials such as Dirac andWeyl semimetals, are also elucidated.AEI/FEDER, (UE) MAT2016-79866-

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

Energy Criterion for the Spectral Stability of Discrete Breathers

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather’s energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials

Nonlinear instabilities of multi-site breathers in Klein–Gordon lattices

In the present work, we explore the possibility of excited breather states in a nonlinear Klein–Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this fundamentally 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 hard (φ4) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the excited breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, excited breather states are observed to be nonlinearly stable

Stability of traveling waves in a driven Frenkel–Kontorova model

In this work we revisit a classical problem of traveling waves in a damped Frenkel–Kontorova lattice driven by a constant external force. We compute these solutions as fixed points of a nonlinear map and obtain the corresponding kinetic relation between the driving force and the velocity of the wave for different values of the damping coefficient. We show that the kinetic curve can become non-monotone at small velocities, due to resonances with linear modes, and also at large velocities where the kinetic relation becomes multivalued. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection. We discuss why the validity of this criterion in the dissipative setting is a rather remarkable feature offering connections to the Hamiltonian variant of the model and of lattice traveling waves more generally. Our stability results are corroborated by direct numerical simulations which also reveal the possible outcomes of dynamical instabilities.AEI/FEDER, (UE) MAT2016-79866-

Moving breather collisions in the Peyrard-Bishop DNA model

We consider collisions of moving breathers (MBs) in the Peyrard-Bishop DNA model. Two identical stationary breathers, sep- arated by a fixed number of pair-bases, are perturbed and begin to move approaching to each other with the same module of velocity. The outcome is strongly dependent of both the velocity of the MBs and the number of pair-bases that initially separates the stationary breathers. Some col- lisions result in the generation of a new stationary trapped breather of larger energy. Other collisions result in the generation of two new MBs. In the DNA molecule, the trapping phenomenon could be part of the complex mechanisms involved in the initiation of the transcription pro- cesses

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