Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice

We study the existence and stability of two-dimensional discrete breathers in a two-dimensionai discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling. Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential （soft or hard）, the interaction potential （attractive or repulsive） and the center of the two-dimensional gap breathers （on a light or a heavy atom）.     

Stable and metastable states and the formation and destruction of breathers in the discrete nonlinear Schrödinger equation

Statistical mechanics explains many localization phenomena of lattices such as the discrete nonlinear Schrödinger equation. However, numerical simulations show that the complete thermalization is rarely achieved. Instead, one observes metastable statistical states that are robust when excited locally. This paper investigates thermodynamically metastable states where the trajectory is confined to some part of the energy shell. The partition function and the entropy are computed with a perturbation method. This method is applicable to stable and metastable states, and it allows us to give approximative analytic expressions for the entropy in the complete thermodynamic state space.     

Effect of Nonlinearity on Scattering Dynamics of Solitary Waves

We discuss the effect of nonlinearity on the scattering dynamics of solitary waves. The pure nth power model with the interaction potential V （Х） = Х^n/n is present, which is a paradigm model in the study of solitary waves. The dependence of the scattering property on nonlinearity is closely related to the topological structures of the solitary waves. Moreover, for one of the four collision types, the rates of energy loss increase with the strength of nonlinearity and would reach 1 at n ≥ 10, which means that the two solitary waves would become of fragments completely after the collision.     

Deterministic escape of a dimer over an anharmonic potential barrier

In the present paper we consider the deterministic escape dynamics of a dimer from a metastable state over an anharmonic potential barrier. The underlying dynamics is conservative and noiseless and thus, the allocated energy has to suffice for barrier crossing. The two particles comprising the dimer are coupled through a spring. Their motion takes place in a two-dimensional plane. Each of the two constituents for itself is unable to escape, but as the outcome of strongly chaotic coupled dynamics the two particles exchange energy in such a way that eventually exit from the domain of attraction may be promoted. We calculate the corresponding critical dimer configuration as the transition state and its associated activation energy vital for barrier crossing. It is found that there exists a bounded region in the parameter space where a fast escape entailed by chaotic dynamics is observed. Interestingly, outside this region the system can show Fermi resonance which, however turns out to impede fast escape.     

Existence and Stability of Compact-Like Discrete Breather in Discrete One-Dimensional Monatomic Chains

Compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. It is proven that compact-like discrete breathers exist not only in soft Ф^4 potential but also in hard Ф^4 potential and K4 chains. The measurements of compact-like discrete breathers＇ core in soft and hard Ф^4 potential are determined by coupling parameter K4, while the measurements of compact-like discrete breathers＇ core in K4 chains are not related to coupling parameter K4. The stabilities of compact-like discrete breathers correlate closely to coupling parameter K4 and the boundary condition of lattice.     

Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line

We introduce an extended nonlinear Schrödinger (ENLS) equation describing the dynamics of modulated waves in a nonlinear discrete electrical transmission line (NLTL) with nonlinear dispersion. We show that this equation admits envelope dark solitary wave with compact support, with width and speed independent of the amplitude, as a solution. Analytical criteria of existence and stability of this solution are derived. In particular, we show that the modulated compact wave may exist in the NLTL depending on the frequency range of the chosen carrier wave, for physically realistic parameters. The stability of compact dark solitary wave is confirmed by numerical simulations of this ENLS equation and the exact equations of the network.     

Existence and Stability of Two-Dimensional Compact-Like Discrete Breathers in Discrete Two-Dimensional Monatomic Square Lattices

Two-dimensional compact-like discrete breathers in discrete two-dimensional monatomic square lattices are investigated by discussing a generalized discrete two-dimensional monatomic model. It is proven that the two- dimensional compact-like discrete breathers exist not only in two-dimensional soft Φ4 potentials but also in hard two-dimensional Φ4 potentials and pure two-dimensional K4 lattices. The measurements of the two-dimensional compact-like discrete breather cores in soft and hard two-dimensional Φ4 potential are determined by coupling parameter K4, while those in pure two-dimensional K4 lattices have no coupling with parameter K4. The stabilities of the two-dimensional compact-like discrete breathers correlate closely to the coupling parameter K4 and the boundary condition of lattices.     

Parametrically Driven Solitons in a Chain of Nonlinear Coupled Pendula with an Impurity

The system consisting of a chain of parametrically driven and damped nonlinear coupled pendula with a mass impurity is studied by means of a discrete version of the envelope function approach. An analogue of the parametrically driven damped nonlinear Schodinger equation with an impurity term is derived from the original lattice equation. Analytical solutions of impurity pinned high-frequency breathers and kinks are obtained. The results show that the mass impurity has striking influence on the high-frequency modes. In addition, we perform numerical simulations, showing that the light mass impurity has a stabilizing effect on the chain. The breathers seeding chaos in the homogeneous chain are pinned on a suitable light impurity to pull the chain from the chaotic state.     

Periodic, Quasiperiodic and Chaotic Discrete Breathers in a Parametrical Driven Two-Dimensional Discrete Klein-Gordon Lattice

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein-Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.     

Multi-site Compact-Like Discrete Breather in Discrete One-Dimensional Monatomic Chains

Multi-site compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. We obtain that the two-site compact-like discrete breathers with codes σ = （0,..., 0, 1, 1, 0,..., 0）and codes σ= （0,..., 0, 1, -1, 0, ..., 0） can exist in discrete one-dimensional monatomic chain with quartic on-site and inter-site potentials. However, the former can only exist in hard quartic on-site potential and cannot exist in soft quartic on-site potential, whereas the latter is just reversed. A11 of the two-site Compact-like discrete breathers with codes σ = （0,..., 0, 1, 1, 0,..., 0） and σ （0,... ,0, 1, -1,0,... ,0} cannot exist in a pure K4 chain.     

Two-Dimensional Discrete Gap Breathers in a Two-Dimensional Diatomic β Fermi--Pasta--Ulam Lattice

We study the existence of two-dimensional discrete breathers in a two-dimensional face-centred square lattice consisting of alternating light and heavy atoms, with nearest-neighbour coupling containing quartic soft or hardnonlinearity. This study is focused on two-dimensional breathers with frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of existence of two-dimensional gap breathers by using the numerical method, the local anharmonicity approximation and the rotating wave approximation. We obtain six types of two-dimensional gap breathers, i.e., symmetric, mirror-symmetric and asymmetric, no matter whether the centre of the breather is on a light or a heavy atom.     

The zone boundary mode in periodic nonlinear electrical lattices

We study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of N inductors and N voltage-dependent capacitors with periodic boundary conditions. The inductances are allowed to alternate, while the capacitors are identical and each have a quadratic dependence on voltage. By block-diagonalizing a 2N×2N Floquet problem, we reduce the question of the stability of the mode to a single Hill’s equation that is analyzed using methods of perturbation theory and averaging. We show that periodicity of the lattice inductances degrades stability, and also show that the instability threshold is proportional to N⁻². Numerical computations validate the perturbative results.     

On Some Classes of New Solutions of Continuous β-FPU Chain

A continuous β-Fermi--Pasta--Ulam (FPU) chain is investigated by using the knowledge of elliptic equation and Jacobian elliptic functions. We obtain the new solutions, two-kink soliton solution, breather solution and breather lattice solution, of the continuous β-FPU chain, besides the kink-soliton solution and chaos solution.     

Quasicontinuum approximation and iterative method for envelope solitons in anharmonic chains

For solitary waves on a monoatomic chain with nearest neighbor interactions the continuum approximation has a limited validity range and exhibits certein mathematical problems. For pulse solitons these problems are overcome by the Quasicontinuum Approach (QCA), and the validity range is considerably extended. We generalize the QCA to oscillatory excitations and derive analytic expressions for bright and dark envelope solitons, limiting ourselves to a polynomial interaction potential with harmonic, cubic and quartic terms. Moreover we describe and apply a numerical iteration procedure in Fourier space in order to take into account discreteness effects in a systematic way. This procedure yields envelope solitons with a width in the order of the lattice constant. In the case of zero velocity these solutions can be compared with intrinsic localized modes derived by other authors. The stability and accuracy of all our solutions are tested by numerical simulations.     

Extremal entropy for a constrained probability density

The state of extremal entropy for a one-dimensional probability density is considered. This density is constrained by fixed values of the first and second moment. The grandcanonical distribution yields the extremum of the entropy within a certain range of values of the moments. A different type of density corresponds to an extremum of entropy when the moments are outside this range. The shape of this density is approximated with the Ritz variation method. The results are applied to the formation of breathers in the discrete nonlinear Schrödinger equation.     

Anomalous decay and breather formation in doped alkali halides

Anomalous decay of doped alkali halides has been ascribed to breather formation in the immediate neighborhood of the impurity. New results support this connection. We report experimental data for NaBr and RbBr crystals showing anomalies in their slow emission decay. These data complete the earlier picture, confirming that the decay anomaly becomes bigger as the host-lattice-anion/cation mass ratio increases. We show the correlation between the decay anomaly and the breather formation as a function of this ratio. By simulating finite-temperature effects (which do not just involve white noise) in the lattice dynamics, we demonstrate that with increasing temperature the breather weakens, as is experimentally observed for the decay anomaly.     

Anharmonic excitations in the FPU and FK models

The properties of vibrational localized (breathers) and traveling (anharmonic phonons) waves are discussed in an infinite, one-dimensional, monoatomic crystal for the Fermi-Pasta-Ulam and Frenkel-Kontorova models. The shooting and finite difference schemes have been implemented to reckon the displacement fields of breathers and anharmonic phonons, respectively. These tools provide localized and traveling waves proving to be indefinitely stable in simulations carried out by solving the equations of motion. The emphasis is laid on the role of the cubic and quartic terms of the anharmonic potential which turn out to oppose and to shore up the restoring force, respectively. The case of vibrational modes arising in an anharmonic crystal subject to a soft phonon induced instability is also addressed. Received 7 November 2001 and Received in final form 5 February 2002 Published online 6 June 2002     

On global asymptotic stability of 2-D discrete systems with state saturation

A criterion for the global asymptotic stability of 2-D discrete systems described by the Roesser model employing state saturation nonlinearities is presented. The criterion is a less restrictive version of an earlier criterion due to Liu and Michel.     

Localized modes in a one-dimensional sphalerite-structure lattice with anharmonicity

The nonlinear localized vibrational modes of a one-dimensional atomic chain with two periodically alternating masses and force constants are analytically investigated using a discrete multiple-scale expansion method. This model simulates a row of atoms in the <1 1 1>-direction of sphalerite, or zinc blende, crystals. Owing to the structural asymmetry, the vibrational amplitude is governed by a perturbed nonlinear Schr?dinger equation instead of the standard one found in one-dimensional lattices with two alternating masses but uniform force constant. Although the stationary localized modes with carrier wavevector at the Brillouin-zone boundary are similar to those of ionic lattices, the moving localized modes with wavevectors within the zone are different owing to the perturbation. The calculation shows that the height of the moving localized modes in this lattice dampens with time. Received 14 May 2001 and Received in final form 12 July 2001     

Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays

In this Letter, we have dealt with the problem of lag synchronization and parameter identification for a class of chaotic neural networks with stochastic perturbation, which involve both the discrete and distributed time-varying delays. By the adaptive feedback technique, several sufficient conditions have been derived to ensure the synchronization of stochastic chaotic neural networks. Moreover, all the connection weight matrices can be estimated while the lag synchronization is achieved in mean square at the same time. The corresponding simulation results are given to show the effectiveness of the proposed method.