We investigate the interplay of nonreciprocity and nonlinearity in a one-dimensional nonlinear Klein-Gordon chain of classical oscillators coupled by asymmetric springs, akin to a mechanical analog of the Hatano-Nelson model with on-site nonlinearity. Using multiple-scale analysis, we show that families of nonlinear skin breathing modes-time-periodic, boundary-localized oscillations-emerge from their linear counterparts at any nonreciprocal strength. We derive an explicit nonlinear frequency shift for these families of nonlinear breathing modes, showing its dependence on amplitude, nonlinearity type, lattice size, and nonreciprocity, and we predict the emergence of genuine skin end breathers at the boundary once their oscillation frequency and higher harmonics enter the spectral gaps of the linear spectrum. Numerical pseudo-arclength continuation confirms full families of solutions for both hardening and softening nonlinearities. Furthermore, the Floquet analysis shows that these modes can be either linearly stable or unstable, with Floquet eigenvectors exhibiting skin localization inherited from the asymmetric couplings. Our results extend the nonlinear non-Hermitian skin effect from stationary modes to intrinsically time-periodic excitations, providing a pathway to engineer and control breathing modes in nonreciprocal mechanical metamaterials.

We uncover subtle and previously unexplored phenomena arising from the interplay of nonlinearity and non-reciprocity in topological mechanical metamaterials. We study a non-reciprocal topological Klein-Gordon chain of asymmetrically coupled nonlinear oscillators, which serves as a minimal mass-spring model capturing the features of several active nonreciprocal metamaterials across mechanical, electronic and acoustic platforms. We demonstrate that continuous families of non-reciprocal edge breathers (NEBs), namely boundary-localized, time-periodic waves, emerge from the linear edge mode as its amplitude increases. Remarkably, despite the absence of chiral or sublattice symmetries, we identify insensitive NEBs whose nonlinear frequency remains fixed to that of the linear edge mode with increasing nonlinearity. Our analysis reveals that the mechanism underlying this insensitivity stems from a competition between mode non-orthogonality and nonlinear interactions, yielding an exponential decay of the NEB nonlinear frequency shift with system size. Crucially, these insensitive NEBs also persist in the strongly nonlinear regime. Our work establishes a novel pathway towards realizing robust nonlinear topological waves in mechanical metamaterials without relying on symmetry-protected nonlinearities.

The time evolution of initially localized wavepackets in the discrete Hatano-Nelson lattice displays a rich dynamical structure shaped by the interplay between dispersion and nonreciprocity. Our analysis reveals a characteristic evolution of the wave-packet center of mass, which undergoes an initial acceleration, subsequently slows down, and ultimately enters a regime of uniform motion, accompanied throughout by exponential amplification of the wave-packet amplitude. To capture this behavior, we develop a continuum approximation that incorporates higher-order dispersive and nonreciprocal effects and provides accurate analytical predictions across all relevant time scales. Building on this framework, we then demonstrate the existence of a non-Hermiticity-induced jump - an abrupt spatial shift of the wave-packet center even in the absence of disorder - and derive its underlying analytical foundation. The analytical predictions are in excellent agreement with direct numerical simulations of the Hatano-Nelson chain. Our results elucidate the interplay between dispersion and nonreciprocity in generating unconventional transport phenomena, and pave the way for controlling wave dynamics in nonreciprocal and non-Hermitian metamaterials.

We demonstrate that wave amplification enables even weak nonlinearities to reshape linear wave-packet transport in nonreciprocal systems. We study the dynamics of bulk Gaussian wave packets in the Hatano–Nelson model with onsite cubic nonlinearity. We show that the interplay between nonlinearity and amplification generates growing frequency shifts that drive the wave packet through three successive dynamical regimes: an early nonlinear-skin regime with coherent propagation, an intermediate wave-mixing regime driven by mode resonances, and a self-trapping regime in which part of the packet localizes while the remainder ballistically spreads along the system favored direction. The crossover time scales are set by the width and average spacing of the eigen-frequency spectrum. Crucially, within the nonlinear-skin regime, we derive analytical predictions for the wave-packet dynamics and show that nonlinearity couples amplification, dispersion, and nonreciprocity, thereby modifying the magnitude of the wave-packet acceleration and introducing an explicit time dependence into its evolution. Focusing nonlinearities suppress the acceleration and cause it to decrease in time, whereas defocusing nonlinearities enhance it and cause it to increase. We further show that nonlinear interactions typically break down the wave packet before the non-Hermitian jump can occur. Our results provide a route toward accurate control of waves in nonreciprocal metamaterials.

We uncover subtle and previously unexplored phenomena arising from the interplay of nonlinearity and nonreciprocity in topological mechanical metamaterials. We study a nonreciprocal topological Klein-Gordon chain of asymmetrically coupled nonlinear oscillators, which serves as a minimal mass-spring model capturing the features of several active nonreciprocal metamaterials across mechanical, electronic, and acoustic platforms. We demonstrate that continuous families of nonreciprocal edge breathers (NEBs), namely boundary-localized, time-periodic waves, emerge from the linear edge mode as its amplitude increases. Remarkably, despite the absence of chiral or sublattice symmetries, we identify insensitive NEBs whose nonlinear frequency remains fixed to that of the linear edge mode with increasing nonlinearity. Our analysis reveals that the mechanism underlying this insensitivity stems from a competition between mode nonorthogonality and nonlinear interactions, yielding an exponential decay of the NEB nonlinear frequency shift with system size. Crucially, these insensitive NEBs also persist in the strongly nonlinear regime. Our work establishes a novel pathway toward realizing robust nonlinear topological waves in mechanical metamaterials without relying on symmetry-protected nonlinearities.

We present a mechanism to generate unidirectional pulse-shaped propagating waves, tamed to exponential growth and dispersion, in active systems with nonreciprocal and nonlinear couplings. In particular, when all bulk modes are exponentially localized at one side of the lattice (skin effect), it is expected that wave dynamics is governed by amplification or decay until reaching the boundaries, even in the presence of dissipation. Our analytical results, and experimental demonstrations in an active electrical transmission line metamaterial, reveal that nonlinearity is a crucial tuning parameter in mediating a delicate interplay between nonreciprocity, dispersion, and dissipation. Consequently, undistorted unidirectional solitonic pulses are supported both for low and high nonreciprocity and pulse amplitude strength. The proposed mechanism facilitates robust pulse propagation in signal processing and energy transmission applications.

We present a method for the computation of the Generalized Alignment Index (GALI), a fast and effective chaos indicator, using a multi-particle approach that avoids variational equations. We show that this approach is robust and accurate by deriving a leading-order error estimation for both the variational (VM) and the multi-particle (MPM) methods, which we validate by performing extensive numerical simulations on two prototypical models: the two degrees of freedom H & eacute;non-Heiles system and the multidimensional /i-Fermi-Pasta-Ulam-Tsingou chain of oscillators. The dependence of the accuracy of the GALI on control parameters such as the renormalization time, the integration time step and the deviation vector size is studied in detail. We test the MPM implemented with double precision accuracy (E approximate to 10-16) and find that it performs reliably for deviation vector sizes d0 approximate to E E1 E1/ E1/2 E1/2, renormalization times a <= <= 1, and relative energy errors Er <= E E1 E1/ E1/2 E1/2. These results hold for systems with many degrees of freedom and demonstrate that the MPM is a robust and efficient method for studying the chaotic dynamics of Hamiltonian systems. Our work makes it possible to explore chaotic dynamics with the GALI in a vast number of systems by eliminating the need for variational equations.

One-dimensional topological acoustic systems offer robust signal transmission and localization in compact spaces. Here, we demonstrate the ability to control both longitudinal and transverse topological waves in a quasi-one-dimensional elastic lattice, composed of an array of repelling magnets connected to elastic cantilevers. The two different topological modes can be separately or simultaneously excited at selected frequencies. Our system enables controlled multimodal propagation and edge localization, owing to its finite size and to the design of longitudinal and transverse topological band gaps within similar frequency ranges. In addition to the robustness of the topological modes to unintentional disorder, we show that disorder can couple degrees of freedom and tailor the energy distribution between longitudinal and transverse waves. This system may enable other studies on the interactions of topological modes useful in sensing and information-processing applications.

Non-Hermitian lattices with non-reciprocal couplings under open boundary conditions are known to possess linear modes exponentially localized on one edge of the chain. This phenomenon, dubbed non-Hermitian skin effect, induces all input waves in the linearized limit of the system to unidirectionally propagate toward the system's preferred boundary. Here we investigate the fate of the non-Hermitian skin effect in the presence of Kerr-type nonlinearity within the well-established Hatano-Nelson lattice model. Our method is to probe the presence of nonlinear stationary modes which are localized at the favored edge, when the Hatano-Nelson model deviates from the linear regime. Based on perturbation theory, we show that families of nonlinear skin modes emerge from the linear ones at any non-reciprocal strength. Our findings reveal that, in the case of focusing nonlinearity, these families of nonlinear skin modes tend to exhibit enhanced localization, bridging the gap between weakly nonlinear modes and the highly nonlinear states (discrete solitons) when approaching the anti-continuum limit with vanishing couplings. Conversely, for defocusing nonlinearity, these nonlinear skin modes tend to become more extended than their linear counterpart. To assess the stability of these solutions, we conduct a linear stability analysis across the entire spectrum of obtained nonlinear modes and also explore representative examples of their evolution dynamics.

In this work, we demonstrate that the synergetic interplay of topology, nonreciprocity and nonlinearity is capable of unprecedented effects. We focus on a nonreciprocal variant of the Su-Shrieffer-Heeger chain with local Kerr nonlinearity. We find a continuous family of non-reciprocal edge solitons (NES) emerging from the topological edge mode, with near-zero energy, in great contrast from their reciprocal counterparts. Analytical results show that this energy decays exponentially towards zero when increasing the lattice size. Consequently, despite the absence of chiral and sublattice symmetries within the system, we obtain zero-energy NES, which are insensitive to growing Kerr nonlinearity. Even more surprising, these zero-energy NES also persist in the strong nonlinear limit. Our work may enable new avenues for the control of nonlinear topological waves without requiring the addition of complex chiral- or sublattice-preserving nonlinearities.

We numerically investigate the characteristics of the long-time dynamics of a single-site wave-packet excita-tion in a disordered and nonlinear Su-Schrieffer-Heeger model. In the linear regime, as the parameters controlling the topology of the system are varied, we show that the transition between two different topological phases is preceded by an anomalous diffusion, in contrast to Anderson localization within these topological phases. In the presence of onsite nonlinearity this feature is lost due to mode-mode interactions. Direct numerical simulations reveal that the characteristics of the asymptotic nonlinear wave-packet spreading are the same across the whole studied parameter space. Our findings underline the importance of mode-mode interactions in nonlinear topological systems, which must be studied in order to define reliable nonlinear topological markers.

We consider a mechanical lattice inspired by the Su-Schrieffer-Heeger model along with cubic Klein-Gordon type nonlinearity. We investigate the long-time dynamics of the nonlinear edge states, which are obtained by nonlinear continuation of topological edge states of the linearized model. Linearly unstable edge states delocalize and lead to chaos and thermalization of the lattice. Linearly stable edge states also reach the same fate, but after a critical strength of perturbation is added to the initial edge state. We show that the thermalized lattice in all these cases shows an effective renormalization of the dispersion relation. Intriguingly, this renormalized dispersion relation displays a unique symmetry, i.e., its square is symmetric about a finite squared frequency, akin to the chiral symmetry of the linearized model.

We study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the disordered discrete nonlinear Schr\"odinger (DDNLS) equation of one (1D) and two (2D) spatial dimensions. We present detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE--$\Lambda$), and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law $t^{\alpha_\Lambda}$ with $\alpha_\Lambda<0$ for both the weak and strong chaos regimes, no crossover to the behavior $\Lambda \propto t^{-1}$ (which is indicative of regular motion) is observed. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites.

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

We introduce a modified version of the disordered Klein-Gordon lattice model, having two parameters for controlling the disorder strength: $D$, which determines the range of the coefficients of the on-site potentials, and $W$, which defines the strength of the nearest-neighbor interactions. We fix $W=4$ and investigate how the properties of the system's normal modes change as we approach its ordered version, i.e. $D\rightarrow 0$. We show that the probability density distribution of the normal modes' frequencies takes a `U'-shaped profile as $D$ decreases. Furthermore, we use two quantities for estimating the modes' spatial extent, the so-called localization volume $V$ (which is related to the mode's second moment) and the mode's participation number $P$. We show that both quantities scale as $\propto D^{-2}$ when $D$ approaches zero and we numerically verify a proportionality relation between them as $V/P \approx 2.6$.

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a "graphene-type" force law and one inspired by Hollomon's law describing "work-hardening" effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles N and the total energy E are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large N, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.

We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density and that the onset of chaos in graphene is slow, becoming evident after more than 104 natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges, the chaoticity of GNRs is stronger than the periodic graphene sheet and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Furthermore, we show that the composition of 12C and 13C carbon isotopes in graphene has a minor impact on its chaotic strength.

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.

Received 24 May 2019DOI:https://doi.org/10.1103/PhysRevE.99.069903©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChaosClassical mechanicsCoupled oscillatorsPhysical SystemsChaotic systemsDynamical systemsHamiltonian systemsHigh dimensional systemsNonlinear wavesTechniquesPhase space methodsTime series analysisNonlinear Dynamics