The dynamics of complex-amplitude norm-preserving lattices of coupled oscillators (original) (raw)

Localized Oscillations in Nonlinear Lattices: Existence and Stability

2005

Numerical methods using homoclinic orbits are applied to study the existence and stability of spatially localized and time-periodic oscillations of 1-dimensional (1D) nonlinear lattices, with linear interaction between nearest neighbors and a quartic on-site potential 4 2 4 1 2 1) (u Ku u V ± = where the (+) sign corresponds to "hard spring" and (-) to "soft spring" models. These localized oscillations-when they are stable under small perturbations-are very important for physical systems, since they seriously affect the energy transport properties of the lattice. We use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α, ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of V(u), we find that the regions of existence and stability for simple breathers of the "hard spring" lattice are considerably larger than those of the "soft spring" system. The variation of the size of the regular region around a stable breather is investigated as the number of particles is increased.

Statics and dynamics of an inhomogeneously nonlinear lattice

Physical Review E, 2006

We introduce an inhomogeneously-nonlinear Schrödinger lattice, featuring a defocusing segment, a focusing segment and a transitional interface between the two. We illustrate that such inhomogeneous settings present vastly different dynamical behavior than the one expected in their homogeneous counterparts in the vicinity of the interface. We analyze the relevant stationary states, as well as their stability by means of perturbation theory and linear stability analysis. We find good agreement with the numerical findings in the vicinity of the anticontinuum limit. For larger values of the coupling, we follow the relevant branches numerically and show that they terminate at values of the coupling strength which are larger for more extended solutions. The dynamical development of relevant instabilities is also monitored in the case of unstable solutions.

Damping and pumping of localized intrinsic modes in nonlinear dynamical lattices

Physical review, 1994

It has been recently demonstrated that dynamical models of nonlinear lattices admit approximate solutions in the form of self-supported intrinsic modes (IM s). In this work, the intensity of the emission of radiation ("phonons") from the oue-dimensional IM is calculated in an analytical approximation for the case of a moderately strong anharmonicity. Contrary to the emission in nonintegrable continuum models, which may be summarized as fusion of several vibrons into a phonon, the emission in the lattice may be described in terms of fission of a vibron into several phonons: as the IM's internal frequency lies above the phonon. band of the lattice, the radiative decay of the IM in the discrete system can be only subharmonic. It is demonstrated that the corresponding lifetime of the IM may be very large. Then, the threshold (minimum) value of the amplitude of an external ac 6eld, necessary to support the IM in a lattice with dissipative losses, is found for the limiting cases of the weak and strong anharmonicity.

Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices

Physical Review Letters, 2000

In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q fi 0, p. Incommensurate analytic SWs with jQj . p͞2 may however appear as "quasistable," as their instability growth rate is of higher order. 42.65.Sf, 45.05. + x, 63.20.Ry A well known and rather spectacular phenomenon occurring in many nonlinear media (e.g., fluids or optical waveguides) is the modulational (Benjamin-Feir) instability (MI), by which a traveling plane wave breaks up into a train of solitary waves (see, e.g., ). It is also well known that wave propagation in many continuous nonlinear media is well described by nonlinear Schrödingertype equations, where the solitary wave trains are described by spatially periodic and stable standing wave (SW) solutions, the so-called cnoidal envelope waves .

On Moving Discrete Modes in Nonlinear Lattices

2008

Results of a comprehensive dynamical analysis are reported for several fundamental species of bright solitons in the one-dimensional lattice modeled by the discrete nonlinear Schrödinger equation with the cubicquintic nonlinearity. Staggered solitons, which were not previously considered in this model, are studied numerically, through the computation of the eigenvalue spectrum for modes of small perturbations, and analytically, by means of the variational approximation. The numerical results confirm the analytical predictions. The mobility of discrete solitons is studied by means of direct simulations, and semianalytically, in the framework of the Peierls-Nabarro barrier, which is introduced in terms of two different concepts, free energy and mapping analysis. It is found that persistently moving localized modes may only be of the unstaggered type.

Ground-state properties of small-size nonlinear dynamical lattices

Physical Review E, 2007

We investigate the ground state of a system of interacting particles in small nonlinear lattices with M ≥ 3 sites, using as a prototypical example the discrete nonlinear Schrödinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays, and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground state (modulational) instability appears to be intimately connected with a non-standard ("double transcritical") type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.

Translationally invariant nonlinear Schrödinger lattices

Nonlinearity, 2006

The persistence of stationary and travelling single-humped localized solutions in the spatial discretizations of the nonlinear Schrödinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions. Another constraint on the nonlinear function is found from the condition that the differential advance-delay equation for travelling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction gives a necessary condition for existence of single-humped travelling solutions. The nonlinear function which admits both reductions defines a fourparameter family of discrete NLS equations which generalizes the integrable Ablowitz-Ladik lattice. Particular travelling solutions of this family of discrete NLS equations are written explicitly.

Dynamics of lattice kinks

Physica D: Nonlinear Phenomena, 2000

We consider a class of Hamiltonian nonlinear wave equations governing a field defined on a spatially discrete one dimensional lattice, with discreteness parameter, d = h −1 , where h > 0 is the lattice spacing. The specific cases we consider in detail are the discrete sine-Gordon (SG) and discrete φ 4 models. For finite d and in the continuum limit (d → ∞) these equations have static kink-like (heteroclinic) states which are stable. In contrast to the continuum case, due to the breaking of Lorentz invariance, discrete kinks cannot be "Lorentz boosted" to obtain traveling discrete kinks. Peyrard and Kruskal pioneered the study of how a kink, initially propagating in the lattice dynamically adjusts in the absence of an available family of traveling kinks. We study in detail the final stages of the discrete kink's evolution during which it is pinned to a specified lattice site (equilibrium position in the Peierls-Nabarro barrier). We find: (i) for d sufficiently large (sufficiently small lattice spacing), the state of the system approaches an asymptotically stable ground state static kink (centered between lattice sites). (ii) for d sufficiently small d < d * the static kink bifurcates to one or more time periodic states. For the discrete φ 4 we have: wobbling kinks which have the same spatial symmetry as the static kink as well as "g-wobblers" and "e-wobblers", which have different spatial symmetry. In the discrete sine-Gordon case, the "ewobbler" has the spatial symmetry of the kink whereas the "g-wobbler" has the opposite one. These time-periodic states may be regarded as a class of discrete breather / topological defect states; they are spatially localized and time periodic oscillations mounted on a static kink background.