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We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns. We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one, two and three dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parametrized families. The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impu- rities while lower bounds, with respect to the kinetic energy, are also discussed for traveling waves in FPU periodic lattices.

International audience; We study the existence of discrete breathers (time-periodic and spatially localized oscillations) in a chain of coupled nonlinear oscillators modelling the breathing of DNA. We consider a modification of the Peyrard-Bishop model introduced by Peyrard et al. [Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys. 35 (2009), 73-89], in which the reclosing of base pairs is hindered by an energy barrier. Using a new kind of continuation from infinity, we prove for weak couplings the existence of large amplitude and low frequency breathers oscillating around a localized equilibrium, for breather frequencies lying outside resonance zones. These results are completed by numerical continuation. For resonant frequencies (with one multiple belonging to the phonon band) we numerically obtain discrete breathers superposed on a small oscillatory tail.

International audience; We consider an infinite chain of particles linearly coupled to their nearest neighbors and subject to an anharmonic local potential. The chain is assumed weakly inhomogeneous, i.e. coupling constants, particle masses and on-site potentials can have small variations along the chain. We look for small amplitude and time-periodic solutions, and, in particular, spatially localized ones (discrete breathers). The problem is reformulated as a nonautonomous recurrence in a space of time-periodic functions, where the dynamics is considered along the discrete spatial coordinate. Generalizing to nonautonomous maps a center manifold theorem previously obtained for infinite-dimensional autonomous maps [44], we show that small amplitude oscillations are determined by finite-dimensional nonautonomous mappings, whose dimension depends on the solutions frequency. We consider the case of two-dimensional reduced mappings, which occur for frequencies close to the edges of the phonon band (computed for the unperturbed homogeneous chain). For an homogeneous chain, the reduced map is autonomous and reversible, and bifurcations of reversible homoclinic orbits or heteroclinic solutions are found for appropriate parameter values. These orbits correspond respectively to discrete breathers for the infinite chain, or "dark" breathers superposed on a spatially extended standing wave. Breather existence is shown in some cases for any value of the coupling constant, which generalizes (for small amplitude solutions) an existence result obtained by MacKay and Aubry at small coupling [57]. For an inhomogeneous chain, the study of the nonautonomous reduced map is in general far more involved. Here, the problem is considered when the chain presents a finite number of defects. For the principal part of the reduced recurrence, using the assumption of weak inhomogeneity, we show that homoclinics to 0 exist when the image of the unstable manifold under a linear transformation (depending on the defect sequence) intersects the stable manifold. This provides a geometrical understanding of tangent bifurcations of discrete breathers commonly observed in classes of systems with impurities as defect strengths are varied. The case of a mass impurity is studied in detail, and our geometrical analysis is successfully compared with direct numerical simulations. In addition, a class of homoclinic orbits is shown to persist for the full reduced mapping and yields a family of discrete breathers with maximal amplitude at the impurity site.

We investigate how nonlinearity and topological disorder affect the energy relaxation of local kicks in coarse-grained network models of proteins. We find that nonlinearity promotes long-range, coherent transfer of substantial energy to specific, functional sites, while depressing transfer to generic locations. Remarkably, transfer can be mediated by the self-localization of discrete breathers at distant locations from the kick, acting as efficient energy-accumulating centers., 4 pages, 3 figures

Discrete Breathers (DBs) are phenomena of energy localization in networks of anharmonic oscillators. Recent literature has shown that DBs might be present at various places in protein structures, along with reports linking DBs explicitly with the functions of some enzymes. I will talk about my work in describing what and why locally happens to anharmonic oscillators, and how that may relate to the larger networks, and, hence, to applications in modelling of protein strucutres and dynamics.

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.

Collisions between moving localized modes (moving breathers) in non- integrable lattices present a rich outcome. In this paper, some features of the interaction of moving breathers in Discrete Nonlinear Schrödinger and Klein- Gordon lattices, together with some plausible explanations, are exposed.

We investigate localization phenomena and stability properties of quasiperiodic oscillations in $N$ degree of freedom Hamiltonian systems and $N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of $N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date., Comment: 16 pages, 8 figures, submitted for publication in Journal of Computational and Applied Mathematics (Elsevier)

The International Workshop on Complex Systems (5º. 2007. Sendai, Japan) Reconstructive transformations of layered silicates as mica muscovite take place at much lower temperatures than expected. A possible explanation is the existence of breathers within the potassium layer. Numerical analysis of a model shows the existence of many different types of breathers with different energies and existence ranges which spectrum coincides approximately with a statistical theory for them. Ministerio de Educacion y Ciencia, Spain, project FIS2004-01183

We analyze behavior of spatially extended network of coupled oscillators. We demonstrate an original method of showing the breather-like behavior by local analysis of oscillation of uncoupled oscillator, and properties of the associated twist map.

We introduce a topology-based nonlinear network model of protein dynamics with the aim of investigating the interplay of spatial disorder and nonlinearity. We show that spontaneous localization of energy occurs generically and is a site-dependent process. Localized modes of nonlinear origin form spontaneously in the stiffest parts of the structure and display site-dependent activation energies. Our results provide a straightforward way for understanding the recently discovered link between protein local stiffness and enzymatic activity. They strongly suggest that nonlinear phenomena may play an important role in enzyme function, allowing for energy storage during the catalytic process., 4 pages, 5 figures. Minor changes

The existence of highly localized multi-site oscillatory structures (discrete multibreathers) in a nonlinear Klein-Gordon chain which is characterized by an inverse dispersion law is proven and their linear stability is investigated. The results are applied in the description of vertical (transverse, off-plane) dust grain motion in dusty plasma crystals, by taking into account the lattice discreteness and the sheath electric and/or magnetic field nonlinearity. Explicit values from experimental plasma discharge experiments are considered. The possibility for the occurrence of multibreathers associated with vertical charged dust grain motion in strongly-coupled dusty plasmas (dust crystals) is thus established. From a fundamental point of view, this study aims at providing a first rigorous investigation of the existence of intrinsic localized modes in Debye crystals and/or dusty plasma crystals and, in fact, suggesting those lattices as model systems for the study of fundamental crystal properties., 12 pages, 8 figures, revtex format

Discrete breathers or intrinsic localized modes are nonlinear localized states that appear in several classical extended systems, such as for instance the Fermi-Paste-Ulam (FPU) model. In order to probe the quantum states that correspond to discrete breathers, we quantize the beta-FPU model using boson quantization rules, retain only number conserving terms, and analyze the two-quanta sector of the model. For both attractive and repulsive nonlinearity, we find the occurrence of biphonons in two forms, on-site and nearest-neighbor site, and analyze their properties. We comment on the use of this model as a minimal model for extended molecular and biomolecular systems. (c) 2006 Elsevier B.V. All rights reserved. Conference on Nonlinear Physics - Condensed Matter, Dynamical Systems and Biophysics, May 30-31, 2005, Inst Henri Poincare, Paris, France

13 pages, 9 figures., We study the structure and stability of discrete breathers (both pinned and mobile) in two-dimensional nonlinear anisotropic Schrödinger lattices. Starting from a set of identical one-dimensional systems we develop the continuation of the localized pulses from the weakly coupled regime (strongly anisotropic) to the homogeneous one (isotropic). Mobile discrete breathers are seen to be a superposition of a localized mobile core and an extended background of two-dimensional nonlinear plane waves. This structure is in agreement with previous results on one-dimensional breather mobility. The study of the stability of both pinned and mobile solutions is performed using standard Floquet analysis. Regimes of quasi-collapse are found for both types of solutions, while another kind of instability (responsible for the discrete breather fission) is found for mobile solutions. The development of such instabilities is studied, examining typical trajectories on the unstable nonlinear manifold., The authors acknowledge F. Falo, Yu. Kivshar, R.S. Mackay and M. Peyrard for sharing thoughts, and pointing out some important references to us. JG-G and LMF are grateful to M. Johansson and B. Malomed for discussions on some issues regarding the “travelling wave” (orthodox) perspective on discrete breathers. Financial support came from MCyT (Projects No. BFM2002 00113 and FIS2005 00337), DGA and BIFI. JG-G acknowledges financial support from the MECyD through a FPU grant. Work at Los Alamos performed under the auspices of the US DoE.

International audience; Standard Kramers theory of chemical reactions involves a coupling with a Langevin thermal bath which intrinsically forbids the possible existence of Discrete Breathers (DBs) (i.e. local modes). However, it is now known that in complex systems, that energy may focus for long time as Discrete Breathers (local mode). In very special systems, targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature. The dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable (chemical expressway). A simple toy model illustrating this idea is reduced to a Rotor weakly coupled to a Morse oscillator (supposed to represent two specific local modes in a complex system) which are appropriately tuned for targeted energy transfer. When the Rotor is initially rotating with a frequency resonant with those of theMorse oscillator at rest, the energy of the Rotor is almost completely transferred to the Morse oscillator and induces chemical dissociation. The periodic oscillations of the Rotor andMorse oscillator remain coherent and their frequencies simultaneously vary, but always remain resonant. This process is analytically described within an integrable approximation. Numerical investigations of this model confirm that in the appropriate conditions, the particle in the Morse oscillator is indeed promptly ejected at infinity with a finite velocity (chemical dissociation) despite some chaotic transient manifesting imperfect integrability.

Pre-pint tomado de Arxiv Discrete breathers, or intrinsic localized modes, are spatially localized, time–periodic, nonlinear excitations that can exist and propagate in systems of coupled dynamical units. Recently, some experiments show the sighting of a form of discrete breather that exist at the atomic scale in a magnetic solid. Other observations of breathers refer to systems such as Josephson–junction arrays, photonic crystals and optical-switching waveguide arrays. All these observations underscore their importance in physical phenomena at all scales. The authors review some of their latest theoretical contributions in the field of classical and quantum breathers, with possible applications to these widely different physical systems and to many other such as DNA, proteins, quantum dots, quantum computing, etc.

Recent papers have considered moving breathers (MBs) in DNA models including long range interaction due to the dipole moments of the hydrogen bonds. We have recalculated the value of the charge transfer when hydrogen bonds stretch using quantum chemical methods which takes into account the whole nucleoside pairs. We explore the consequences of this value on the properties of MBs, including the range of frequencies for which they exist and their effective masses. They are able to travel through bending points with fairly large curvatures provided that their kinetic energy is larger than a minimum energy which depends on the curvature. These energies and the corresponding velocities are also calculated in function of the curvature., 4 pages and 4 figures

The DNA molecule is modeled by a parabola embedded chain with long-range interactions between twisted base pair dipoles. A mechanism for bubble generation is presented and investigated in two different configurations. Using random normally distributed initial conditions to simulate thermal fluctuations, a relationship between bubble generation, twist and curvature is established. An analytical approach supports the numerical results., 7 pages, 8 figures. Accepted for Phys. Rev. E (in press)

A bent chain of coupled Morse oscillators with long-range dispersive interaction is considered. Moving localized excitations may be trapped in the bending region. Thus chain geometry acts like an impurity. An energy funneling effect is observed in the case of random initial conditions., 6 pages, 12 figures. Submitted to Physical Review E, Oct. 13, 2003

Reconstructive transformations in layered silicates need a high tem- perature in order to be observed. However, very recently, some systems have been found where transformation can be studied at temperatures 600 C below the lowest experimental results previously reported, including sol-gel methods. We explore the possible relation with the existence of intrinsic localized modes, known as discrete breathers. We construct a model for nonlinear vibrations within the cation layer, obtain their parameters and calculate them numerically, obtaining their energies. Their statistics shows that although there are far less breathers than phonons, there are much more above the activation energy, being therefore a good candidate to explain the reconstructive transformations at low temperature., Comment: 27 pages, 11 figures

We study the interaction of moving localized oscillations with a local inhomogeneity in a discrete nonlinear Hamiltonian system. We conjecture that resonance with a static nonlinear localized oscillation centered at the local inhomogeneity is a necessary condition for observing the trapping phenomenon. Analytic calculations and numerical simulations agree well with our hypothesis.

Whereas there exists a mathematical proof for one--site breathers stability, and an unpublished one for two--sites breathers, the methods for determining the stability properties of multibreathers rely in numerical computation of the Floquet multipliers or in the weak nonlinearity approximation leading to discrete non--linear Schr\"odinger equations. Here we present a set of multibreather stability theorems (MST) that provides with a simple method to determine multibreathers stability in Klein--Gordon systems. These theorems are based in the application of degenerate perturbation theory to Aubry's band theory. We illustrate them with several examples., Comment: 22 pages, 4 figures

A vacancy defect is described by a Frenkel--Kontorova model with a discommensuration. This vacancy can migrate when interacts with a moving breather. We establish that the width of the interaction potential must be larger than a threshold value in order that the vacancy can move forward. This value is related to the existence of a breather centred at the particles adjacent to the vacancy., 11 pages, 10 figures

We study the properties of moving breathers in a bent DNA-related model with short range interaction, due to the stacking of the base pairs, and long range interaction, due to the finite dipole moment of the bonds within each base pair. We show that the movement of a breather is hindered by the bending of the chain as a particle in a potential barrier. We have also found that the behaviour of moving breathers in an homogeneous bent chain and in a straight chain with a small impurity is qualitatively equivalent.

Moving breathers is a means of transmitting information in DNA. We study the existence and properties of moving breathers in a DNA model with short range interaction, due to the stacking of the base pairs, and long range interaction, due to the finite dipole moment of the bond within each base pair. In our study, we have found that mobile breathers exist for a wide range of the parameter values, and the mobility of these breathers is hindered by the long range interaction. This fact is manifested by: (a) an increase of the effective mass of the breather with the dipole–dipole coupling parameter; (b) a poor quality of the movement when the dipole–dipole interaction increases; and (c) the existence of a threshold value of the dipole–dipole coupling above which the breather is not movable. An analytical formula for the boundaries of the regions where breathers are movable is calculated. Concretely, for each value of the breather frequency, it can be obtained the maximum value of the dipole–dipole coupling parameter and the maximum and minimum values of the stacking coupling parameter where breathers are movable. Numerical simulations show that, although the necessary conditions for the mobility are fulfilled, breathers are not always movable. Finally, the value of the dipole–dipole coupling constant is obtained through quantum chemical calculations. They show that the value of the coupling constant is small enough to allow a good mobility of breathers. European Commission under the RTN project LOCNET, HPRN-CT-1999-00163

We present a study of nonlinear localized excitations called discrete breathers in a superconducting array. These localized solutions were recently observed in Josephson-junction ladder arrays by two different experimental groups. We review the experiments made by Trias et al. We report the detection of different single-site and multi-site breather states and study the dynamics when changing the array bias current. By changing the temperature we can control the value of the damping (the Stewart-McCumber parameter) in the array, thus allowing an experimental study at different array parameters. We propose a simple DC circuit model to understand most of the features of the detected states. We have also compared this model and the experiments with simulations of the dynamics of the array. We show that the study of the resonances in the ladder and the use of harmonic balance techniques allow for understanding of most of the numerical results. We have computed existence diagrams of breather solutions in our arrays, found resonant localized solutions and described the localized states in terms of vortex and antivortex motion., Accepted in Physica D

Most of the studies on mathematical models of DNA are limited to next neighbor interaction. However, the coupling between base pairs is thought to be caused by dipole interaction, and, when the DNA strand is bent, the distances between base pairs become shorter, therefore the interactions with distant base pairs have to be taken into account. In this paper we analyze the existence and stability of breathers, i.e., localized oscillations in a simple model of bent DNA with long-range dipole interaction. Breathers have been suggested as precursors of the denaturation bubble.