Nonlinear Lattice Model Research Articles

Thermal conductivities of one-dimensional anharmonic/nonlinear lattices: renormalized phonons and effective phonon theory

Heat transport in low-dimensional systems has attracted enormous attention from both theoretical and experimental aspects due to its significance to the perception of fundamental energy transport theory and its potential applications in the emerging field of phononics: manipulating heat flow with electronic anologs. We consider the heat conduction of one-dimensional nonlinear lattice models. The energy carriers responsible for the heat transport have been identified as the renormalized phonons. Within the framework of renormalized phonons, a phenomenological theory, effective phonon theory, has been developed to explain the heat transport in general one-dimensional nonlinear lattices. With the help of numerical simulations, it has been verified that this effective phonon theory is able to predict the scaling exponents of temperature-dependent thermal conductivities quantitatively and consistently.

1D PHONON SCATTERING BY STABLE SPATIALLY INHOMOGENEOUS EQUILIBRIUM SOLUTION IN NONLINEAR DISCRETE KLEIN–GORDON MODEL: EXACT ANALYSIS

Nonlinear Discrete Klein–Gordon (NDKG) model is a nonlinear lattice model used to investigate different nonlinear phenomena raised in many physical contexts. In the present study we use this model to observe linear phonon scattering by stable spatially inhomogeneous equilibrium solution in the piece-wise linear (PWL) version (asymmetric double well) of nonlinearity instead of the conventional cubic or other types of nonlinearities. Considering one-site symmetric solution in such a system we are able to construct the exact expressions for the transmittance (T) and reflectance using the transfer matrix formalism. From the condition of perfect transmission (T = 1, hence reflectance, R = 0) in the elastic scattering process, we observe in our model, such a situation happens to appear at the threshold of a localized mode at the band-edge of the extended eigen modes. The advantage of the PWL version of nonlinearity is that all the results derived are exact.

Multiresonance of energy transport and absence of heat pump in a force-driven lattice

Energy transport control in low dimensional nanoscale systems has attracted much attention in recent years. In this paper, we investigate the energy transport properties of the Frenkel-Kontorova lattice subject to a periodic driving force, in particular, the resonance behavior of the energy current by varying the external driving frequency. It is discovered that, in certain parameter ranges, multiple resonance peaks, instead of a single resonance, emerge. By comparing the nonlinear lattice model with a harmonic chain, we unravel the underlying physical mechanism for such a resonance phenomenon. Other parameter dependencies of the resonance behavior are examined as well. Finally, we demonstrate that heat pumping is actually absent in this force-driven model.

Existence of steady-state solutions in a nonlinear photonic lattice model

Many careful experimental observations to nonlinear photonic lattice model have been constructed. In this paper, we use the principle of variational method, mountain pass lemma, fixed point method to develop an existence theorem for the steady-state solutions of a nonlinear photonic lattice model describing the propagation of a light wave in a photo-refractive crystal is established, which demonstrates that there is an amount of continuous energy that allows the existence of steady-state solutions. Our results provide a theoretical principles for a variety of experiments and research on photonic lattices and crystals. Finally, it is straightforward to see that the applicability of the present to constructing arbitrarily small energy solutions is also guaranteed.

Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide

We study the propagation of quasidiscrete microwave solitons in a nonlinear left-handed coplanar waveguide coupled with split-ring resonators. By considering the relevant transmission line analog, we derive a nonlinear lattice model which is studied analytically by means of a quasidiscrete approximation. We derive a nonlinear Schrödinger equation, and find that the system supports bright envelope soliton solutions in a relatively wide subinterval of the left-handed frequency band. We perform systematic numerical simulations, in the framework of the nonlinear lattice model, to study the propagation properties of the quasidiscrete microwave solitons. Our numerical findings are in good agreement with the analytical predictions, and suggest that the predicted structures are quite robust and may be observed in experiments.

Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments

This paper is a first attempt to derive a qualitatively simple model coupling the dynamics of a charged biopolymer and its diffuse cloud of counterions. We consider here the case of a single actin filament. A zig-zag chain model introduced by Zolotaryuk et al [28] is used to represent the actin helix, and calibrated using experimental data on the stiffness constant of actin. Starting from the continuum drift-diffusion model describing counterion dynamics, we derive a discrete damped diffusion equation for the quantity of ionic charges in a one-dimensional grid along actin. The actin and ionic cloud models are coupled via electrostatic effects. Numerical simulations of the coupled system show that mechanical waves propagating along the polymer can generate charge density waves with intensities in the $pA$ range, in agreement with experimental measurements of ionic currents along actin.

Elastic scattering of traveling phonons by discrete breathers as a time-periodic scatterer in the one-dimensional discrete nonlinear Schrödinger model: Exact analysis

Discrete nonlinear Schrödinger model is a nonlinear lattice model used to investigate different nonlinear phenomena arising in many physical contexts. In this work we used this model to observe linear traveling phonon scattering by time-periodic discrete breathers in the piecewise smooth (PWS) version of nonlinearity. In one-dimensional system the single-channel scattering is found to be elastic (with the incoming and outgoing fluxes of energy being equal to each other). Considering one-site symmetric breather solution we are able to calculate the exact expression for the transmission coefficient (T) or the ratio of the transmitted to incident flux of energy in such a system using transfer-matrix formalism. From the condition of perfect transmission ( T=1 , and hence reflection coefficient R=0 ) or the ratio of reflected to incident flux of energy for elastic scattering is zero, our observation shows that perfect transmission happens to appear at the threshold of a localized mode, which occurs at the band edge of the extended eigenmodes (plane waves). We have also presented the results obtained from the condition of perfect reflection. The advantage of using the PWS version of nonlinearity in the model is that all the results derived are exact. Numerical simulations complement our results.

Reflection Wave Suppression in Frenkel-Kontrova Model Simulation: Analysis Based on the GKYP Lemma

The Frenkel-Kontorova model is a nonlinear lattice model to represent extremely complicated behaviors such as a dislocation in crystal and DNA structures. This model is given as the dynamical syste...

Heat conduction in the nonlinear response regime: Scaling, boundary jumps, and negative differential thermal resistance

We report a numerical study on heat conduction in one-dimensional homogeneous lattices in both the linear and the nonlinear response regime, with a comparison among three prototypical nonlinear lattice models. In the nonlinear response regime, negative differential thermal resistance (NDTR) can occur in both the Frenkel-Kontorova model and the phi4 model. In the Fermi-Pasta-Ulam- beta model, however, only positive differential thermal resistance can be observed, as shown by a monotonous power-law dependence of the heat flux on the applied temperature difference. In general, it was found that NDTR can occur if there is nonlinearity in the onsite potential of the lattice model. It was also found that the regime of NDTR becomes smaller as the system size increases, and eventually vanishes in the thermodynamic limit. For the phi4 model, a phenomenological description of the size-induced crossover from the existence to the nonexistence of a NDTR regime is provided.

Nonlinear breathing modes at a defect site in DNA

Molecular-dynamics simulations of a normal DNA duplex show that breathing events typically occur on the microsecond time scale. This paper analyzes a 12 base pairs DNA duplex containing the "rogue" base difluorotoluene (F) in place of a thymine base (T), for which the breathing events occur on the nanosecond time scale. Starting from a nonlinear Klein-Gordon lattice model and adding noise and damping, we obtain a mesoscopic model of the DNA duplex close to that observed in experiments and all-atom molecular dynamics simulations. The mesoscopic model is calibrated to data from the all-atom molecular dynamics package AMBER for a variety of twist angles of the DNA duplex. Defects are considered in the interchain interactions as well as in the along-chain interactions. This paper also discusses the role of the fluctuation-dissipation relations in the derivation of reduced (mesoscopic) models, the differences between the potential of mean force and the potential energies used in Klein-Gordon lattices, and how breathing can be viewed as competition between the along-chain elastic energy and the interchain binding energy.

Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures

We study nonlinear waves in a two-layered imperfectly bonded structure using a nonlinear lattice model. The key element of the model is an anharmonic chain of oscillating dipoles, which can be viewed as a basic lattice analog of a one-dimensional macroscopic waveguide. Long nonlinear longitudinal waves in a layered lattice with a soft middle (or bonding) layer are governed by a system of coupled Boussinesq-type equations. For this system we find conservation laws and show that pure solitary waves, which exist in a single equation and can exist in the coupled system in the symmetric case, are structurally unstable and are replaced with generalized solitary waves.

Nonlinear Analysis of the Dynamics of DNA Breathing

The base pairs that encode the genetic information in DNA show large amplitude localized excitations called DNA breathing. We discuss the experimental observations of this phenomenon and its theoretical analysis. Starting from a model introduced to study the thermal denaturation of DNA, we show that it can qualitatively describe DNA breathing but is quantitatively not satisfactory. We show how the model can be modified to be quantitatively correct. This defines a nonlinear lattice model, which is interesting in itself because it has nonlinear localized excitations, forming a new class of discrete breather.

Correction of multiple nonlinear resonances in storage rings

The correct implementation of the nonlinear lattice model is crucial to achieving the design performance in storage rings. We describe here a method for the simultaneous correction of multiple nonlinear resonances based on local resonance measurements and numerical fits of the sextupole components. This method has been applied for the simultaneous correction of two sextupole resonances excited in the Diamond storage ring. The local correction of these resonances has been achieved with unprecedented precision. We also point out that this method has the potential to lead to an effective reconstruction of the local sextupole component errors around the whole ring circumference.

Visualizing intrinsic localized modes with a nonlinear micromechanical array

Micromechanical cantilever arrays provide the opportunity to visualize the nonlinear excitations of a discrete nonlinear system in real time. Both stationary and moving localized nonlinear excitations can be produced either by driving the system at a frequency outside the plane wave spectrum or by driving the system at a frequency within the small amplitude dispersion curve range. To see these modes the tips of the cantilevers are imaged on a 1D CCD camera. The brightness of the image depends on the oscillation amplitude of the cantilever, so that a distribution of amplitudes in the array can be recorded as a function of position and time. Both the stationary and traveling excitations have been successfully simulated using a nonlinear lumped element lattice model. The former ILM can appear in any size lattice while the latter requires a low density of modes for the formation of smoothly running excitation.

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.

Anomalous Heat Conduction in Three-Dimensional Nonlinear Lattices

Heat conduction in three-dimenisional nonlinear lattice models is studied using nonequilibrium molecular dynamics simulations. We employ the FPU model, in which there exists a nonlinearity in the interaction of biquadratic form. It is confirmed that the thermal conductivity, the ratio of the energy flux to the temperature gradient, diverges in systems up to 128x128x256 lattice sites. This size corresponds to nanoscopic to mesoscopic scales of several tens of nanometers. From these results, we conjecture that the energy transport in insulators with perfect crystalline order exhibits anomalous behavior. The effects of lattice structure, random impurities, and natural length in interactions are also examined. We find that face-centered cubic (fcc) lattices display stronger divergence than simple cubic lattices. When impurity sites of infinitely large mass, which are hence fixed, are randomly distributed, such divergence vanishes.

Discrete breathers in two-dimensional nonlinear lattices

Properties of discrete breathers in two- and three-dimensional nonlinear lattice models are discussed with special emphasis on possible structures of localization in two-dimensional case. In contrast to the extensive studies addressing the various features of discrete breathers in one-dimensional lattices, less studies have been done in regard to the higher dimensional breathers. To begin with a brief survey of existing results for discrete breathers in higher dimensional lattices, we attempt to look into the possible structures of higher dimensional breathers. Based on the survey, we address the following topics concerning the discrete breathers in two-dimensional lattices: types of localized structures; chaotic breathers; quasi-continuum approximation; discrete breathers versus solitons in continuum.

High-order-mode soliton structures in two-dimensional lattices with defocusing nonlinearity

While fundamental-mode discrete solitons have been demonstrated with both self-focusing and defocusing nonlinearity, high-order-mode localized states in waveguide lattices have been studied thus far only for the self-focusing case. In this paper, the existence and stability regimes of dipole, quadrupole, and vortex soliton structures in two-dimensional lattices induced with a defocusing nonlinearity are examined by the theoretical and numerical analysis of a generic envelope nonlinear lattice model. In particular, we find that the stability of such high-order-mode solitons is quite different from that with self-focusing nonlinearity. As a simple example, a dipole ("twisted") mode soliton with adjacent excited sites which may be stable in the focusing case becomes unstable in the defocusing regime. Our results may be relevant to other two-dimensional defocusing periodic nonlinear systems such as Bose-Einstein condensates with a positive scattering length trapped in optical lattices.

Non-linear lattice models: complex dynamics, pattern formation and aspects of chaos

In the present paper we intend to discuss the question of pattern formation and complex dynamics developed in some lattice models. Dynamical systems are characterized by their coherent and chaotic natures. In this paper, we report non-linear lattice models which are good candidates to illustrate the fascinating complex dynamics occurring in some physical systems. One of the proposed models is devoted to the formation of localized structures in a two-dimensional lattice. Instability process of localized structures is examined in detail. The instability process is mostly related to the phase transformation in crystalline solids. On the basis of a two-dimensional lattice model involving non-linear and competing interactions the formation and dynamics of elastic domains and twin boundaries are investigated. The emphasis is placed especially on the instability mechanisms of a strain band and modulated strain structure with respect to the transverse perturbations then producing localized structures on the lattice. The long-time evolution of localized modes is also studied using asymptotic perturbative methods and an asymptotic equation is then deduced in the framework in the quasicontinuum approximation. Next, we propose a non-linear analysis in the vicinity of a critical point of the phonon-dispersion branch. From the non-linear analysis, we deduce, in the semidiscrete approach by using a multiscale technique, an amplitude equation of the Ginzburg–Landau type. The mechanism of self-generated non-linear localized modes in the two-dimensional lattice beyond the instability process or near the critical region of the phonon dispersion is numerically investigated. The numerical simulations exhibit non-trivial localized patterns. The second study deals with the soliton motion on a damped driven one-dimensional lattice. The lattice model is made of a one-dimensional chain equipped with rotatory molecules. The problem of soliton dynamics under the influence of discreteness effects, damping and time-dependent applied field allows us to show a transition to chaos of the soliton movement. Numerical simulations performed directly on the discrete system show a particularly rich dynamics of the lattice model. The last model we want to propose for illustration of the dynamics of non-linear systems is a one-dimensional lattice model with long-range interactions. In this part of the paper we are interested in the non-local description of the discrete system in the framework of the quasicontinuum approach. The idea is to consider a one-to-one correspondence between functions of discrete arguments and a class of analytic functions. The procedure allows us to use the same representation of the Lagrangian for all cases: discrete and analytic. The equation of motion deduced in the quasicontinuum representation is similar to that of a non-local model of elasticity including a local non-linear term. A perturbative asymptotic multiscale technique leads to an asymptotic equation for the long-time evolution of the non-linear wave. Numerical simulations are performed on the discrete system for different ranges of non-local action showing the propagation of discrete kink in the lattice.

Positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.