We study nonlinear excitations propagating in a hexagonal layer which is a model for the cation layer of silicates. We consider their properties in the frequency-momentum or $ω-k$ representation, extending the theory on pterobreathers in their moving frame for the first time to two dimensions. It can also be easily extended to three dimensions. Exact traveling waves in the $ω-k$ representation are within {\em resonant} planes, each plane corresponding in the moving frame to a single frequency. These frequencies are integer multiples of a frequency called the fundamental frequency. A breather is within a resonant plane called the breather plane and has a single frequency in the moving frame. The intersection of the resonant planes with the phonon surfaces produce co-traveling wings with a small set of frequencies. The traveling waves obtained by perturbing the system consist of a breather and a soliton traveling together and are quasi-exact. These traveling waves can be used as seeds to obtain exact traveling waves, also formed by a breather and a soliton. The wings do exist but they are usually very small.
We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions…
We study discrete breathers in prototypical nonlinear oscillator networks subjected to non-harmonic zero-mean periodic excitations. We show how the generation of stationary and moving discrete breathers are optimally controlled by solely varying the impulse transmitted by the periodic excitations, while keeping constant the excitation's amplitude and period. Our theoretical and numerical results show that the enhancer effect of increasing values of the excitation's impulse, in the sense of facilitating the generation of stationary and moving breathers, is due to a correlative increase of the b…
Interaction of moving discrete breathers with vacancies
本文采用具有非线性相互作用势的Frenkel-Kontorova模型来描述晶体中的空位缺陷。根据最近的数值结果[Cuevas等人，Phys. Lett. A 315，364（2003）]，当空位与移动的呼吸器相互作用时，空位可以迁移。我们更深入地研究了运动呼吸子与单空位和双空位相互作用所引起的唯象。我们发现，空位的流动性是强烈相关的存在和稳定性的固定呼吸中心的粒子附近的空位，我们现在将称为空位呼吸。
In the present work we revisit the existence, stability and dynamical properties of moving discrete breathers in $β$-FPU lattices. On the existence side, we propose a numerical procedure, based on a continuation along a sequence of velocities, that allows to systematically construct breathers traveling more than one lattice site per period. On the stability side, we explore the stability spectrum of the obtained waveforms via Floquet analysis and connect it to the energy-frequency bifurcation diagrams. We illustrate in this context examples of the energy being a multivalued function of the fre…
In this paper we consider a 2D hexagonal crystal lattice model first proposed by Marin, Eilbeck and Russell in 1998. We perform a detailed numerical study of nonlinear propagating localized modes, that is, propagating discrete breathers and kinks. The original model is extended to allow for arbitrary atomic interactions, and to allow atoms to travel out of the unit cell. A new on-site potential is considered with a periodic smooth function with hexagonal symmetry. We are able to confirm the existence of long-lived propagating discrete breathers. Our simulations show that, as they evolve, breat…
The formation and evolution of stationary and moving breather solutions in (2+1)-dimensional O(3) nonlinear $σ$-model are investigated. The analytical form of oscillating solutions for (2+1)-dimensional sine-Gordon equation, which evolve to periodic (breather) radially symmetric solutions is determined. On the basis of the found solutions by adding the rotations to the A3-field vector in isotopic space of S^2, the solutions for the O(3) nonlinear $σ$-model are obtained. By numerical study of the solutions dynamics their stability in a stationary and a moving state for quite a long time (45000 …
We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrodinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find a…
Discrete time quantum walks are unitary maps defined on the Hilbert space of coupled two-level systems. We study the dynamics of excitations in a nonlinear discrete time quantum walk, whose fine-tuned linear counterpart has a flat band structure. The linear counterpart is, therefore, lacking transport, with exact solutions being compactly localized. A solitary entity of the nonlinear walk moving at velocity $v$ would therefore not suffer from resonances with small amplitude plane waves with identical phase velocity, due to the absence of the latter. That solitary excitation would also have to …
We review research on the role of nonlinear coherent phenomena (e.g breathers and kinks) in the formation linear decorations in mica crystal. The work is based on a new model for the motion of the mica hexagonal K layer, which allows displacement of the atoms from the unit cell. With a simple piece-wise polynomial inter-particle potential, we verify the existence of localized long-lived breathers in an idealized lattice at 0K. Moreover, our model allows us to observe long-lived localized kinks. We study the interactions of such localized modes along a lattice direction, and in addition demonst…
We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present i…
One-dimensional chain of pointwise particles harmonically coupled with nearest neighbors and placed in six-order polynomial on-site potentials is considered. Power of the energy source in the form of single ac driven particles is calculated numerically for different amplitudes $A$ and frequencies $ω$ within the linear phonon band. The results for the on-site potentials with hard and soft nonlinearity types are compared. For the hard-type nonlinearity, it is shown that when the driving frequency is close to (far from) the {\em upper} edge of the phonon band, the power of the energy source norma…
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation and its dynamical properties are specified. Processes of soliton complex formation are studied analytically and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons. It is shown that moving bound soliton complexes with internal structu…
The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example…
Experimental results describing random, uni-directional, long crested, water waves over non-uniform bathymetry confirm the formation of stable coherent wave packages traveling with almost uniform group velocity. The waves are generated with JONSWAP spectrum for various steepness, height and constant period. A set of statistical procedures were applied to the experimental data, including the space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random field and traveling over …
We study the dynamics of self-trapping in Bose-Einstein condensates (BECs) loaded in deep optical lattices with Gaussian initial conditions, when the dynamics is well described by the Discrete Nonlinear Schrödinger Equation (DNLS). In the literature an approximate dynamical phase diagram based on a variational approach was introduced to distinguish different dynamical regimes: diffusion, self-trapping and moving breathers. However, we find that the actual DNLS dynamics shows a completely different diagram than the variational prediction. We numerically calculate a detailed dynamical phase diag…
It was recently demonstrated that two-dimensional Townes solitons (TSs) in two-component systems with cubic self-focusing, which are normally made unstable by the critical collapse, can be stabilized by linear spin-orbit coupling (SOC), in Bose-Einstein condensates and optics alike. We demonstrate that one-dimensional TSs, realized as optical spatial solitons in a planar dual-core waveguide with dominant quintic self-focusing, may be stabilized by SOC-like terms emulated by obliquity of the coupling between cores of the waveguide. Thus, SOC offers a universal mechanism for the stabilization of…
