Nonlinear Lattice Research Articles

Positive and negative integrable lattice hierarchies: Conservation laws and [formula omitted]fold Darboux transformations

Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and N−fold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual 1−fold DT, this kind of N−fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive N−fold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.

Linearized Wave Turbulence Convergence Results for Three-Wave Systems

We consider stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh-Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh-Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters.

Dynamics of the Berezinskii–Kosterlitz–Thouless transition in a photon fluid

In addition to enhancing confinement, restricting optical systems to two dimensions gives rise to new photonic states, modified transport and distinct nonlinear effects. Here we explore these properties in combination and experimentally demonstrate a Berezinskii–Kosterlitz–Thouless phase transition in a nonlinear photonic lattice. In this topological transition, vortices are created in pairs and then unbind, changing the dynamics from that of a photonic fluid to that of a plasma-like gas of free (topological) charges. We explicitly measure the number and correlation properties of free vortices, for both repulsive and attractive interactions (the photonic equivalent of ferromagnetic and antiferromagnetic conditions), and confirm the traditional thermodynamics of the Berezinskii–Kosterlitz–Thouless transition. We also suggest a purely fluid interpretation, in which vortices are nucleated by inhomogeneous flow and driven by seeded instability. The results are fundamental to optical hydrodynamics and can impact two-dimensional photonic devices if temperature and interactions are not controlled properly. A topological transition in a nonlinear photonic lattice results in new vortex dynamics and a change from photonic fluid behaviour to that of a plasma-like gas.

On the generation and propagation of solitary waves in integrable and nonintegrable nonlinear lattices

We investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems. We begin by discussing the kinetic and potential energy of a solitary wave in these systems and show that under certain circumstances the kinetic and potential energy profiles in these systems (i.e., their spatial distribution) look reasonably close to each other. While this and other features, such as the connection between the amplitude and the total energy of the wave, bear similarities between the two models, there are also notable differences, such as the width of the wave. We then study the dynamical behavior of these systems in response to an initial velocity impulse. For the Toda lattice, we do so by employing the inverse scattering transform, and we obtain analytically the ratio between the energy of the resulting solitary wave and the energy of the impulse, as a function of the impulse velocity; we then compare the dynamics of the Toda system to that of the Hertz system, for which the corresponding quantities are obtained through numerical simulations. In the latter system, we obtain a universality in the fraction of the energy stored in the resulting solitary traveling wave irrespectively of the size of the impulse. This fraction turns out to only depend on the nonlinear exponent. Finally, we investigate the relation between the velocity of the resulting solitary wave and the velocity of the impulse. In particular, we provide an alternative proof for the numerical scaling rule of Hertz-type systems.

Synchronization of stochastic lattice equations

In this paper we consider a system of two coupled nonlinear lattice stochastic equations driven by additive white noise processes. We prove the master slave synchronization of the components of the coupled system, namely, for $$t\rightarrow \infty $$ the solution of one of the subsystems (the slave component) converges to the values of a Lipschitz continuous function of the other component, the master component. To establish this kind of synchronization we will prove the existence of an exponentially attracting random invariant manifold for the coupled system.

Energy super-diffusion in 1d deterministic nonlinear lattices with broken standard momentum

The property of total momentum conservation is a key issue in determining the energy diffusion behavior for 1d nonlinear lattices. The super-diffusion of energy has been found for 1d momentum conserving nonlinear lattices with the only exception of 1d coupled rotator model. However, for all the other 1d momentum non-conserving nonlinear lattices studied so far, the energy diffusion is normal. Here we investigate the energy diffusion in a 1d nonlinear lattice model with inverse couplings. For the standard definition of momentum, this 1d inverse coupling model does not preserve the total momentum while it exhibits energy super-diffusion behavior. In particular, with a parity transformation, this 1d inverse coupling model can be mapped into the well-known 1d FPU-β model although they have different phonon dispersion relations. In contrary to the 1d FPU-β model where the long-wave length phonons are responsible for the super-diffusion behavior, the short-wave length phonons contribute to the super-diffusion of energy in the 1d inverse coupling model.

Magnetic response of a highly nonlinear soliton lattice in a monoaxial chiral helimagnet

We present the theory of nonlinear ac magnetic response for the highly nonlinear regime of the chiral soliton lattice. Increasing of the dc magnetic field perpendicularly to the chiral axis results in crossover inside the phase, when nearly isolated $2\pi$-kinks are partitioned by vast ferromagnetic domains. Assuming that each of the kink reacts independently from the other ones to the external ac field, we demonstrate that internal deformations of such a kink give rise to the nonlinear response.

Two-dimensional discrete breathers in fcc metals

The interest in discrete breathers (DB), i.e. a time-periodic and spatially localized vibrational mode in a defect free nonlinear lattice, is related to their ability to localize vibrational energy of the order of several eV. In the present work, for the first time, a systematic study of eight nonlinear vibrational modes localized in one spatial dimension and delocalized in the two other dimensions is performed by means of molecular dynamics simulations in defect-free single crystals of fcc metals (Al, Cu and Ni). For this goal, the standard embedded atom method potentials are employed. Calculations are performed at a zero temperature in three-dimensional computational cells. Excitation of DBs takes place via displacement of the specified atoms from their equilibrium lattice sites according to the patterns corresponding to the delocalized nonlinear vibrational modes (DNVMs) found earlier for a two-dimensional triangular lattice. It is revealed that only four out of the eight studied DNVMs can support stable two-dimensional DBs with the lifetimes in the range of 24–47 ps and accumulate vibrational energy of the order of 2 eV per atom. All excited DBs are characterized by a hard type of nonlinearity, i.e. the frequency increases with the increasing oscillation amplitude. Obtained results indicate the presence of a wide variety of nonlinear, spatially localized vibrational modes in three-dimensional lattices of fcc metals.

The Lefever–Lejeune nonlinear lattice: Convergence dynamics and the structure of equilibrium states

We consider the Lefever–Lejeune nonlinear lattice, a spatially discrete propagation-inhibition model describing the growth of vegetation densities in dry-lands. We analytically identify parametric regimes distinguishing between decay (associated with spatial extinction of vegetation patches) and potentially non-trivial time-asymptotics. To gain insight on the convergence dynamics, a stability analysis of spatially uniform states is performed, revealing the existence of a threshold for the discretization parameter which depends on the lattice parameters, below which their destabilization occurs and spatially non-uniform equilibrium states may emerge. Direct numerical simulations justified that the analytical stability criteria and parametric thresholds effectively describe the above transition dynamics and revealed the rich structure of the equilibrium set. Connections with the continuous sibling Lefever–Lejeune partial differential equation are also discussed.

Reduced Dilation-Erosion Perceptron for Binary Classification

Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed by the application of a hard-limiter function for binary classification tasks. A DEP classifier can be trained using a convex-concave procedure along with the minimization of the hinge loss function. As a lattice computing model, the DEP classifier assumes the feature and class spaces are partially ordered sets. In many practical situations, however, there is no natural ordering for the feature patterns. Using concepts from multi-valued mathematical morphology, this paper introduces the reduced dilation-erosion (r-DEP) classifier. An r-DEP classifier is obtained by endowing the feature space with an appropriate reduced ordering. Such reduced ordering can be determined using two approaches: one based on an ensemble of support vector classifiers (SVCs) with different kernels and the other based on a bagging of similar SVCs trained using different samples of the training set. Using several binary classification datasets from the OpenML repository, the ensemble and bagging r-DEP classifiers yielded mean higher balanced accuracy scores than the linear, polynomial, and radial basis function (RBF) SVCs as well as their ensemble and a bagging of RBF SVCs.

Rotor interactional effects on aerodynamic and noise characteristics of a small multirotor unmanned aerial vehicle

Small scale unmanned aerial vehicles using multirotor propulsion systems have received considerable attention for a wide range of military and commercial applications in recent years. In the multirotor configuration, the rotor interaction phenomenon occurs severely because the rotors are located in close proximity to each other. Therefore, the separation distance between the adjacent rotor tips has a strong effect on the wake structures and flow fields, which consequently play an important role in determining the aerodynamic performance and noise level of the multirotor vehicle. In the present study, numerical simulations of a quadcopter under hover flight conditions are conducted to investigate the mutual rotor-to-rotor interactional effects on the aerodynamic performance, wake structures, and sound pressure level using the nonlinear vortex lattice method with the vortex particle method and acoustic analogy based on Farassat’s formulation 1A. Calculations for the multirotor configurations with different separation distances show that the average thrust force decreases significantly and force fluctuation is found to increase dramatically as the rotor spacing gets smaller. In addition, the wake geometry and induced flow structure behind the rotor tend to be radially dragged down toward the center of the vehicle due to the existence of the adjoining rotor, which consequently results in strong wake-to-wake interaction and the formation of asymmetric wake structures although the multirotor operates under the hovering condition. It is also observed that unsteady loading introduced by rotor interaction leads to a considerable increase in the sound pressure level, particularly the normal direction of the rotor plane.

Vibrations of nonlinear elastic lattices: low- and high-frequency dynamic models, internal resonances and modes coupling.

We aim to study how the interplay between the effects of nonlinearity and heterogeneity can influence on the distribution and localization of energy in discrete lattice-type structures. As the classical example, vibrations of a cubically nonlinear elastic lattice are considered. In contrast with many other authors, who dealt with infinite and periodic lattices, we examine a finite-size model. Supposing the length of the lattice to be much larger than the distance between the particles, continuous macroscopic equations suitable to describe both low- and high-frequency motions are derived. Acoustic and optical vibrations are studied asymptotically by the method of multiple time scales. For numerical simulations, the Runge-Kutta fourth-order method is employed. Internal resonances and energy exchange between the vibrating modes are predicted and analysed. It is shown that the decrease in the number of particles restricts energy transfers to higher-order modes and prevents the equipartition of energy between all degrees of freedom. The conditions for a possible reduction in the original nonlinear system are also discussed.

A classification algorithm for integrable two-dimensional lattices via Lie—Rinehart algebras

In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic equations of arbitrarily high order integrable in the Darboux sense. Darboux integrablity admits a remarkable algebraic interpretation: the Lie-Rinehart algebras related to both characteristic directions corresponding to the reduced system of the hyperbolic equations have to be of finite dimension. A classification algorithm based on the properties of the characteristic algebra is discussed. Some classification results are presented.

Spherically localized discrete breathers in bcc metals V and Nb

Discrete breathers (DBs), also called intrinsic localized modes (ILMs), are spatially localized, large-amplitude vibrational modes in nonlinear lattices. Rod-like discrete breathers (DBs) have been reported in fcc Ni and bcc Nb by Haas et al. in 2011 based on molecular dynamics simulations. Here we use a general approach to find new type DBs in bcc V and Nb. In this approach, we firstly find the lattice symmetry dictated exact solutions to the equations of atomic motion in the form of delocalized nonlinear vibrational modes (DNVMs). Secondly, a localizing function with spherical symmetry is imposed over the DNVMs. Parameter of the localizing function is chosen such that the obtained DB has a long lifetime. The results presented in this work demonstrate that pure metals can support a variety of DBs. Interatomic potentials have a strong effect on the DB lifetime. Maximal DB lifetime in V is two orders of magnitude larger than that in Nb. The results of this study are interesting for the theory of DBs, and also they will help to better understand the impact of DBs on the physical properties of metals.

Low Reynolds number effects on aerodynamic loads of a small scale wind turbine

Small-scale or scaled-down wind turbines for model experiments mostly operate in low-Reynolds-number flow. The nonlinear variations of aerodynamic coefficients with respect to the angle of attack caused by viscous effects and laminar boundary layer separation affect the wind turbine performance under these conditions. Although the vortex lattice method (VLM) is an efficient way to predict rotor performance, it tends to suffer from numerical error because nonlinear aerodynamic characteristics cannot be considered. In this study, the nonlinear vortex lattice method (NVLM) is adopted to compute the aerodynamic loads of two small-scale wind turbines. This method involves a sectional airfoil look-up table and vortex strength correction and can be applied to a wide range of operating conditions. The simulations of TU Delft and NTNU wind turbines are conducted to validate the prediction capability of numerical models by comparing predictions with the measurements. It was found that the overall results from the NVLM simulation are more accurate than the VLM results, which implies that the nonlinear aerodynamic characteristics associated with low-Reynolds-number flow should be considered to accurately assess the aerodynamic performance of small-sized wind-turbines, particularly at the low tip speed ratio at which the rotor blade may experience flow separation and dynamic stall.

Design of resonant structures in resin matrix to mitigate the blast wave with a very wide frequency range

In this work, resonant structures (RSs) are embedded in the resin matrix to form the micro-scale artificial composite materials to mitigate the blast wave with a very wide frequency range (BWR). The propagation of stress waves in the resin and composite materials is described, and the composite materials exhibit stronger blast wave attenuation characteristic compared with the pure resin material. The attenuation mechanism of the composite materials is explained in detail through the absorption, storage and conversion of impact energy. In addition, the influences of materials of the RSs on the performances of the composite materials are analyzed, and the RS is redesigned to further improve the attenuation effect of the composite material. Equivalent model of the composite material is first proposed and established based on the weakly nonlinear lattice system (WNLS). At the same time, artificial tree algorithm is applied to design its spring stiffness parameters. Based on the WNLS, a three-dimensional composite material plate structure is built to mitigate the overpressure of blast wave at the macro-scale. Compared with traditional materials, the composite material exhibits superior attenuation effect and greater lightweight.

Chaotic wave-packet spreading in two-dimensional disordered nonlinear lattices.

We reveal the generic characteristics of wave-packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schrödinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as t^{a_{m}} with a_{m}≈1/5 (1/3) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S. Flach, Chem. Phys. 375, 548 (2010)CMPHC20301-010410.1016/j.chemphys.2010.02.022]; (b) chaos persists, but its strength decreases in time t since the finite-time maximum Lyapunov exponent Λ decays as Λ∝t^{α_{Λ}}, with α_{Λ}≈-0.37 (-0.46) for the weak (strong) chaos case; and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattice's excited part, which induces the wave packet's thermalization. We also propose a dimension-independent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained α_{Λ} values.

Aerodynamic Modelling of Unmanned Aerial System through Nonlinear Vortex Lattice Method, Computational Fluid Dynamics and Experimental Validation - Application to the UAS-S45 Bàlaam: Part 1.

This paper describes a methodology to predict the aerodynamic behaviour of an Unmanned Aerial System. Aircraft design and flight dynamics modelling are mainly concerned with aerodynamics, and thus its estimation requires a high level of accuracy. The work presented here shows a new non-linear formulation of the classical Vortex Lattice Method and a comparison between this methodology and an experimental analysis. The new non-linear Vortex Lattice Method was performed by calculating the viscous forces from the strip theory, and the forces generated by the vortex rings from the vortex lifting law. The experimental analysis was performed on a reduced scale wing in a low speed wind tunnel. The obtained results were also compared to those obtained from semi-empirical methods programmed using DATCOM and our Fderivatives new in-house codes. The results have indicated the accuracy of the new formulation and showed that an aerodynamic model obtained with the aerodynamic coefficients predicted with this method could be useful for flight dynamics estimation.

Heat transport in one dimension

We review the state of the art of the problem of heat conduction in one dimensional nonlinear lattices. The peculiar role of finite size and time corrections to the predictions of the hydrodynamic theory is discussed. The emerging scenario indicates that when dealing with systems, whose spatial size is comparable with the mean-free path of peculiar nonlinear excitations, hydrodynamic predictions at leading order are no more predictive. We can conjecture that one should take into account estimates of subleading contributions, which could play a major role in some regions of the parameter space in ‘small’ systems.

Monitoring anharmonic phonon transport across interfaces in one-dimensional lattice chains.

Modeling thermal transport through interfaces has been one of the most challenging problems in nanoscale heat transfer. Although continuous theoretical efforts have been made, there has been no consensus on how to rigorously incorporate temperature effect and anharmonicity. In this paper, we adopt the self-consistent anharmonic phonon concept for nonlinear lattices to investigate phonon propagation within the materials as well as across interfaces based on equilibrium molecular dynamics simulations. Based on linear response theory, we propose an efficient method to calculate the frequency-dependent transmission coefficient in a nonlinear lattice. The transmission spectrum is extracted directly from velocity correlations, which naturally includes anharmonic effects. Phonon renormalization at finite temperature can also be easily handled using the proposed method. Our results are consistent with the atomistic Green's function method at the limit of weak anharmonicity. For nonlinear lattices under high temperatures, the anharmonicity is found to increase the cutoff frequency of the transmission coefficient due to phonon renormalization. Further analysis shows that the anharmonicity also promotes interfacial thermal conductance by causing the redistribution of the spectral flux of the excited vibrational waves during their propogation.