Nonlinear Klein-Gordon Chain Research Articles

Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 1: Phase locked traveling breathers

In the present study, we analyze a special type of phase-locked, compact-like, moving breather solution emerging in damped, strongly nonlinear Klein-Gordon chains subjected to parametric traveling-wave excitation. Given the strongly anharmonic nature of the system under consideration, multi-scale analysis of various nonlinear wave phenomena emerging in this model becomes a highly nontrivial task. In this work, we derive the two distinct analytical approaches that enable the analytical characterization of moving, damped-driven breather solutions. As is evident from the results of the present study, the moving breather solutions forming in the damped-driven lattice under consideration are unidirectional. Moreover, their dynamics is fully controlled by the parameters of the applied, traveling wave-like, parametric excitation. These special properties of strongly localized solutions provide the new means for active spatial-temporal control of these strongly localized wave structures.

Universal route to thermalization in weakly-nonlinear one-dimensional chains

We apply Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider $\alpha$ and $\beta$ Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We demonstrate that resonances are responsible for the irreversible transfer of energy among the Fourier modes. We predict that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity. Our methodology is not limited to only these systems and can be applied to the case of a finite number of modes, such as in the original FPUT experiment, or to the thermodynamic limit, i.e. when the number of modes approach infinity. In the latter limit, we perform state of the art numerical simulations and show that the results are consistent with theoretical predictions. We suggest that the route to thermalization, based only on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

Thermalization in the discrete nonlinear Klein-Gordon chain in the wave-turbulence framework

We study the time of equipartition, Teq, of energy in the one-dimensional Discrete Nonlinear Klein-Gordon (DNKG) equation in the framework of the Wave Turbulence (WT) theory. We discuss the applicability of the WT theory and show how this approach can explain qualitatively the route to thermalization and the scaling of the equipartition time as a function of the nonlinear parameter ϵ, defined as the ratio between the nonlinear and linear part of the Hamiltonian. Two scaling laws, and , for different degrees of nonlinearity are explained in terms of four-wave or six-wave processes in the WT theory. The predictions are verified with extensive numerical simulations varying the system size and the degree of nonlinearity.

Multibreather stability in discrete Klein–Gordon equations: Beyond nearest neighbor interactions

Abstract We present results on multibreather stability in one-dimensional nonlinear Klein–Gordon chains. Our analysis is based on Aubryʼs band theory and perturbation theory. First, we provide an alternative proof of the stability of multibreathers in a chain with nearest neighbor interactions only. Then, we extend our analysis to the case of interactions with up to three neighbors. For next-nearest neighbor and third-nearest neighbor interactions we also extend the theory to study the stability properties of recently found multibreathers that have nonstandard phase shifts (not equal to 0 or π).

The notion of persistence applied to breathers in thermal equilibrium

We study the thermal equilibrium of nonlinear Klein–Gordon chains at the limit of small coupling (anticontinuum limit). We show that the persistence distribution associated to the local energy density is a useful tool to study the statistical distribution of the so-called thermal breathers, mainly when the equilibrium is characterized by long-lived pinned excitations; in that case, the distribution of persistence intervals turns out to be a power law. We demonstrate also that this generic behaviour has a counterpart in the power spectra, where the high-frequencies domains nicely collapse if properly rescaled. These results are also compared to nonlinear Klein–Gordon chains with a soft nonlinearity, for which the thermal breathers are rather mobile entities. Finally, we discuss the possibility of a breather-induced anomalous diffusion law, and show that despite a strong slowing down of the energy diffusion, there are numerical evidences for a normal asymptotic diffusion mechanism, but with exceptionally small diffusion coefficients.

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.

Statistical mechanics of nonlinear Klein–Gordon chains: the φ8-chain and the Gaussian double-well model

We study the statistical mechanics of two models of nonlinear Klein–Gordon chain: the ‘ φ 8-chain’ with the single-site potential v( y)= a( y 2−1) 4, and the Gaussian double-well model with v(y)= 1 2 ky 2+b exp(− 1 2 cy 2) where a, b, c and k are positive constants. The thermodynamics of the classical chains is investigated by the transfer-integral equation technique and the pseudo-Schrödinger equation approximation. The results for the heat capacity, the displacement correlation function, and the wave-vector-dependent susceptibility are compared with those of the familiar φ 4-chain. The partition functions of the quantum chains are calculated by the low-coupling effective potential method. The effects of quantum fluctuations on the low-temperature heat capacity are examined.

Moving kinks and nanopterons in the nonlinear Klein–Gordon lattice

We study moving topological solitons (kinks and antikinks) in the nonlinear Klein–Gordon chain. These solitons are shown to exist with both monotonic (non-oscillating) and oscillating asymptotics (tails). Using the pseudo-spectral method, the (anti)kink solutions with oscillating background (so-called nanopterons) are found as travelling waves of permanent profile propagating with constant velocity. Each of these solutions may be considered as a bound state of an (anti)kink with a background nonlinear periodic wave, so that the wave “pushes” the (anti)kink over the Peierls–Nabarro barrier. The stability of these bound states is confirmed numerically. Travelling-wave solutions of permanent profile are shown to exist depending on the convexity of the on-site (substrate) potential. The set of velocities at which the (anti)kinks with monotonic asymptotics propagate freely is calculated. We also find moving non-oscillating (anti)kink profiles with higher topological charges, each of which appears to be the bound state of (anti)kinks with lower topological charge (∣ Q∣=1).

Moving lattice kinks and pulses: an inverse method.

We develop a general mapping from given kink or pulse shaped traveling-wave solutions including their velocity to the equations of motion on one-dimensional lattices which support these solutions. We apply this mapping-by definition an inverse method-to acoustic solitons in chains with nonlinear intersite interactions, nonlinear Klein-Gordon chains, reaction-diffusion equations, and discrete nonlinear Schrödinger systems. Potential functions can be found in a unique way provided the pulse shape is reflection symmetric and pulse and kink shapes are at least C2 functions. For kinks we discuss the relation of our results to the problem of a Peierls-Nabarro potential and continuous symmetries. We then generalize our method to higher dimensional lattices for reaction-diffusion systems. We find that increasing also the number of components easily allows for moving solutions.

Multiscale analysis of discrete nonlinear evolution equations

The method of multiscale analysis is constructed for dicrete systems of evolution equations for which the problem is that of the far behavior of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave resonant interaction system and a discrete modified Volterra model.

Kinks in the Klein-Gordon model with anharmonic interactions: a variational approach

We consider the nonlinear Klein-Gordon chain including anharmonic interatomic interactions. We use a direct perturbation method and a variational approach to obtain the approximate kink solutions of the model in the continuum limit taking the ø 4 and sine-Gordon chains as particular examples. The kink critical velocity due to the cubic anharmonicity is found analytically. Some physically important parameters of the kink, such as its effective mass and energy, are calculated based on the approximate solution and the results are checked by numerical simulations. It is demonstrated that the variational approach yields very accurate results for the kink parameters.

Propagation and stability of kinks in driven and damped nonlinear Klein-Gordon chains.

We consider the propagation of kinks in an elastic chain in a bistable or multistable potential under the action of a driving force. Each element of the chain is subject to a damping force proportional to its velocity. We show that both the propagation velocity of the kinks as a function of the driving field, and the kink width as a function of propagation velocity, are determined by characteristic functions which depend only on the form of the potential. These functions can be found by considering a single particle moving in the upside-down potential of the chain. The general properties of these functions are studied and illustrated by several examples. The stability of these driven kinks is discussed. Interestingly we find in addition to the expected discrete localized eigenmodes a two-dimensional continuum of oscillatory modes with a localized envelope.