7 research outputs found
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
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-
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
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