Introduction: Nonlinear localized modes

Heat transport in long-ranged Fermi-Pasta-Ulam-Tsingou-type models

We perform a detailed study of heat transport in one-dimensional long-ranged anharmonic oscillator systems, such as the long-ranged Fermi-Pasta-Ulam-Tsingou model. For these systems, the long-ranged anharmonic potential decays with distance as a power law, controlled by an exponent δ≥0. For such a nonintegrable model, one of the recent results that has captured quite some attention is the puzzling ballisticlike transport observed for δ=2, reminiscent of integrable systems. Here, we first employ the reverse nonequilibrium molecular dynamics simulations to look closely at the δ=2 transport in three long-ranged models and point out a few problematic issues with this simulation method. Next, we examine the process of energy relaxation, and find that relaxation can be appreciably slow for δ=2 in some situations. We invoke the concept of nonlinear localized modes of excitation, also known as discrete breathers, and demonstrate that the slow relaxation and the ballisticlike transport properties can be consistently explained in terms of a novel depinning of the discrete breathers that makes them highly mobile at δ=2. Finally, in the presence of quartic pinning potentials we find that the long-ranged model exhibits Fourier (diffusive) transport at δ=2, as one would expect from short-ranged interacting systems with broken momentum conservation. Such a diffusive regime is not observed for harmonic pinning.

A coupled "AB" system: Rogue waves and modulation instabilities

Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.

Distributed dynamical computation in neural circuits with propagating coherent activity patterns

Activity in neural circuits is spatiotemporally organized. Its spatial organization consists of multiple, localized coherent patterns, or patchy clusters. These patterns propagate across the circuits over time. This type of collective behavior has ubiquitously been observed, both in spontaneous activity and evoked responses; its function, however, has remained unclear. We construct a spatially extended, spiking neural circuit that generates emergent spatiotemporal activity patterns, thereby capturing some of the complexities of the patterns observed empirically. We elucidate what kind of fundamental function these patterns can serve by showing how they process information. As self-sustained objects, localized coherent patterns can signal information by propagating across the neural circuit. Computational operations occur when these emergent patterns interact, or collide with each other. The ongoing behaviors of these patterns naturally embody both distributed, parallel computation and cascaded logical operations. Such distributed computations enable the system to work in an inherently flexible and efficient way. Our work leads us to propose that propagating coherent activity patterns are the underlying primitives with which neural circuits carry out distributed dynamical computation.

The brain processes information with extraordinary efficiency, and can perform feats such as effortlessly recognizing objects from among thousands of possibilities within a fraction of a second. This is accomplished because the brain represents and processes information in a distributed fashion and in a dynamical way. This processing is manifested in spatiotemporal neural activity patterns of great complexities within the brain. Here, we construct a spiking neural circuit that can reproduce some of the complexities, which are evident in terms of multiple wave patterns with interactions between each other. We show that their dynamics can support propagating pattern-based computation; spiking wave patterns signal information by propagating across neural circuits, and computational operations occur when they collide with each other. Such dynamical computation contrasts sharply with that done by static and physically fixed logic gates operating in other computing machines such as computers. Moreover, we elucidate that the collective dynamics of multiple, interacting wave patterns enable computation processing implemented in a fundamentally distributed and parallel manner in the neural circuit.

Discrete breathers in hexagonal dusty plasma lattices

The occurrence of single-site or multisite localized vibrational modes, also called discrete breathers, in two-dimensional hexagonal dusty plasma lattices is investigated. The system is described by a Klein-Gordon hexagonal lattice characterized by a negative coupling parameter epsilon in account of its inverse dispersive behavior. A theoretical analysis is performed in order to establish the possibility of existence of single as well as three-site discrete breathers in such systems. The study is complemented by a numerical investigation based on experimentally provided potential forms. This investigation shows that a dusty plasma lattice can support single-site discrete breathers, while three-site in phase breathers could exist if specific conditions, about the intergrain interaction strength, would hold. On the other hand, out of phase and vortex three-site breathers cannot be supported since they are highly unstable.

Two-dimensional map for a curved Fermi-Pasta-Ulam chain

A two-dimensional map is derived from the model of a curved Fermi-Pasta-Ulam (FPU) chain which supports exact discrete breather (DB) solutions with frequencies lying outside the linear spectrum or phonon band. The stability of the equilibrium points of the two-dimensional map is examined and the nature of the trajectories is numerically studied. The map displays regular orbits and commensurate states in the phase space for different choices of curvature strength of the FPU chain. The homoclinic map orbits are attributed to the stationary DB solutions of the lattice system.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Existence of multisite intrinsic localized modes in one-dimensional Debye crystals

The existence of highly localized multisite 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 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.

Dynamics of a curved Fermi-Pasta-Ulam chain: effects of geometry, long-range interaction, and nonlinear dispersion

We study the dynamics of the bent Fermi-Pasta-Ulam (FPU) chain, incorporating the complicated effects of geometry, long-range interactions, as well as nonlinear dispersion. Within the rotating wave approximation, we obtain several exact discrete breather (DB) solutions, such as the odd-parity and even-parity discrete breathers, compactlike discrete breathers and moving discrete breathers for various geometries of the chain. In presence of long-range nonlinear dispersive interactions, we show that DBs exist in the discrete curved lattice for next-nearest-neighbor interactions as well. For all neighbors interactions, we treat the problem in the long-wavelength (continuum) and weakly nonlinear limit of the system and obtain exact static breather solutions and large-amplitude, traveling kink-soliton solutions. The curved FPU chain also admits finite amplitude discrete nonlinear sinusoidal wave solutions with short commensurate as well as incommensurate wavelengths. The usefulness of these solutions for energy localization and transport in various physical systems are discussed.

Breathing dissipative solitons in three-component reaction-diffusion system

We investigate the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. A change of the time constants of the inhibitors can lead to a destabilization of the stationary solution. The special case we are interested in is that the breathing mode becomes unstable first and the stationary dissipative soliton undergoes a bifurcation from a stationary to a "breathing" state. This situation is analyzed performing a two-time-scale expansion in the vicinity of the bifurcation point thereby obtaining the corresponding amplitude equation. Also numerical simulations are carried out showing good agreement with the analytical predictions.

Mode-locking of mobile discrete breathers

We study numerically synchronization phenomena of mobile discrete breathers in dissipative nonlinear lattices periodically forced. When varying the driving intensity, the breather velocity generically locks at rational multiples of the driving frequency. In most cases, the locking plateau coincides with the linear stability domain of the resonant mobile breather and desynchronization occurs by the regular appearance of type-I intermittencies. However, some plateaus also show chaotic mobile breathers with locked velocity in the locking region. The addition of a small subharmonic driving tames the locked chaotic solution and enhances the stability of resonant mobile breathers.

Statistical mechanics of general discrete nonlinear Schrödinger models: localization transition and its relevance for Klein-Gordon lattices

We extend earlier work [Phys. Rev. Lett. 84, 3740 (2000)]] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrödinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physically motivated by the desire to describe situations with an excitation threshold for creation of localized excitations, as well as by recent work suggesting noncubic DNLS models to describe Bose-Einstein condensates in deep optical lattices, taking into account the effective condensate dimensionality. Considering ensembles of initial conditions with given values of the two conserved quantities, norm and Hamiltonian, we calculate analytically the boundary of the "normal" Gibbsian regime corresponding to infinite temperature, and perform numerical simulations to illuminate the nature of the localization dynamics outside this regime for various cases. Furthermore, we show quantitatively how this DNLS localization transition manifests itself for small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled anharmonic oscillators (in which energy is the only conserved quantity), and determine conditions for the existence of persistent energy localization over large time scales.

Nonintegrable Schrodinger discrete breathers

In an extensive numerical investigation of nonintegrable translational motion of discrete breathers in nonlinear Schrödinger lattices, we have used a regularized Newton algorithm to continue these solutions from the limit of the integrable Ablowitz-Ladik lattice. These solutions are shown to be a superposition of a localized moving core and an excited extended state (background) to which the localized moving pulse is spatially asymptotic. The background is a linear combination of small amplitude nonlinear resonant plane waves and it plays an essential role in the energy balance governing the translational motion of the localized core. Perturbative collective variable theory predictions are critically analyzed in the light of the numerical results.

Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger equation, we find discrete vortex solitons with various values of the topological charge S. Stability regions for the vortices with S=0,1,3 are investigated. The S=2 vortex is unstable and may spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of microresonators.

Breather statics and dynamics in Klein-Gordon chains with a bend

In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also examine dynamical effects including the scattering of mobile localized modes (discrete breathers) off of such a geometric structure. The potential outcomes of such numerical experiments (including transmission, trapping within the bend as well as reflection) are highlighted and qualitatively explained. Such models are of interest both theoretically in understanding the interplay of breathers with curvature, but also practically in simple models of photonic crystals or of bent chains of DNA.

Breather lattice and its stabilization for the modified Korteweg-de Vries equation

We obtain an exact solution for the breather lattice solution of the modified Korteweg-de Vries equation. Numerical simulation of the breather lattice demonstrates its instability due to the breather-breather interaction. However, such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms.