Stability of traveling waves in a driven Frenkel–Kontorova model

In this work we revisit a classical problem of traveling waves in a damped Frenkel–Kontorova lattice driven by a constant external force. We compute these solutions as fixed points of a nonlinear map and obtain the corresponding kinetic relation between the driving force and the velocity of the wave for different values of the damping coefficient. We show that the kinetic curve can become non-monotone at small velocities, due to resonances with linear modes, and also at large velocities where the kinetic relation becomes multivalued. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection. We discuss why the validity of this criterion in the dissipative setting is a rather remarkable feature offering connections to the Hamiltonian variant of the model and of lattice traveling waves more generally. Our stability results are corroborated by direct numerical simulations which also reveal the possible outcomes of dynamical instabilities.AEI/FEDER, (UE) MAT2016-79866-

New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain

We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simula- tions suggest that they are probably unstable

On the driven Frenkel-Kontorova model: I. Uniform sliding states and dynamical domains of different particle densities

The dynamical behavior of a harmonic chain in a spatially periodic potential (Frenkel-Kontorova model, discrete sine-Gordon equation) under the influence of an external force and a velocity proportional damping is investigated. We do this at zero temperature for long chains in a regime where inertia and damping as well as the nearest-neighbor interaction and the potential are of the same order. There are two types of regular sliding states: Uniform sliding states, which are periodic solutions where all particles perform the same motion shifted in time, and nonuniform sliding states, which are quasi-periodic solutions where the system forms patterns of domains of different uniform sliding states. We discuss the properties of this kind of pattern formation and derive equations of motion for the slowly varying average particle density and velocity. To observe these dynamical domains we suggest experiments with a discrete ring of at least fifty Josephson junctions.Comment: Written in RevTeX, 9 figures in PostScrip

Dry Friction in the Frenkel-Kontorova-Tomlinson Model: Dynamical Properties

Wearless friction is investigated in a simple mechanical model called Frenkel-Kontorova-Tomlinson model. We have introduced this model in [Phys. Rev. B, Vol. 53, 7539 (1996)] where the static friction has already been considered. Here the model is treated for constant sliding speed. The kinetic friction is calculated numerically as well as analytically. As a function of the sliding velocity it shows many structures which can be understood by varies kinds of phonon resonances (normal, superharmonic and parametric) caused by the so-called "washboard wave". For increasing interaction strength the regular motion becomes chaotic (fluid-sliding state). The fluid sliding state is mainly determined by the density of decay channels of m washboard waves into n phonons. We also find strong bistabilities and coherent motions with superimposed dark envelope solitons which interact nondestructively.Comment: Written in RevTeX, figures in PostScript, appears in Z. Phys.

Dissipative phase solitons in semiconductor lasers

We experimentally demonstrate the existence of non dispersive solitary waves associated with a 2$\pi$ phase rotation in a strongly multimode ring semiconductor laser with coherent forcing. Similarly to Bloch domain walls, such structures host a chiral charge. The numerical simulations based on a set of effective Maxwell-Bloch equations support the experimental evidence that only one sign of chiral charge is stable, which strongly affects the motion of the phase solitons. Furthermore, the reduction of the model to a modified Ginzburg Landau equation with forcing demonstrates the generality of these phenomena and exposes the impact of the lack of parity symmetry in propagative optical systems.Comment: 5 pages, 5 figure

Modulation instability gain and localized waves by modified Frenkel-Kontorova model of higher order nonlinearity

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic-quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To calculate the modulation instability gain, we use the linearizing scheme. Particular attention is given to the impact of the higher nonlinear term on the modulation instability. Following that, full numerical integration was performed to identify modulated wave patterns, as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a result, for driving amplitudes above the supratransmission threshold, the solitonic bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright solitonic wave turns into a chaotic solitonic wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves