We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities the localized modes it supports. present first example an oscillatory instability dark solitons, and analyze how may occur for solitons discrete Schr\"odinger generalized Ablowitz-Ladik equations.

We show that the discrete nonlinear Schrödinger (DNLS) equation exhibits exact solutions which are quasiperiodic in time and localized space if ratio between nonlinearity linear hopping constant is large enough. These breather solutions, also exist for a generalized DNLS with on-site nonlinearities of arbitrary positive power, can be constructed by continuation from anticontinuous limit (i.e. zero hopping) where two (or more) sites oscillating incommensurate frequencies. By numerical limit,...

We consider a model for two-dimensional kagome lattice with defocusing nonlinearity, and show that families of localized discrete solitons may bifurcate from linear modes the flat band zero power threshold. Each family such fundamental nonlinear corresponds to unique configuration in strong-nonlinearity limit. By choosing well-tuned dynamical perturbations, small-amplitude, strongly solutions different be switched into each other, as well moved between positions. In window small power,...

We investigate the long-time evolution of weakly perturbed single-site breathers (localized stationary states) in discrete nonlinear Schrodinger equation. The perturbations we consider correspond to time-periodic solutions linearized equations around breather, and can be either (i) spatially localized or (ii) extended. For case (i), which corresponds excitation an internal mode find that interaction between breather its always leads a slow growth amplitude frequency. In (ii), corresponding...

Electron localization and optical transmission in one-dimensional systems with two components distributed according to the Rudin-Shapiro sequence are investigated. The nature of eigenstates diagonal tight-binding model is studied by making use nonlinear recurrence relations satisfied associated transfer matrices their traces. It shown that wave functions display a wide range features going from weak exponential localization. Numerical computations lead conjecture property generic....

We address the issue of mobility localized modes in two-dimensional nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes, e.g., discrete spatial solitons a tight-binding approximation optical waveguide arrays made from photorefractive crystals. discuss numerically obtained exact stationary solutions and their stability, focusing on three different solution families peaks at one, two, four neighboring sites, respectively. When varying power, there is repeated exchange...

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. provide empirical arguments the existence of square summable invariant tori in random nonlinear Schrödinger equation, appearing a finite probability given initial condition sufficiently small norm. Numerical support fat Cantor set conditions generating almost periodic oscillations is obtained by analyzing i) sets recurrent trajectories over successively larger time...

A model equation governing the amplitude of electric field in an array coupled optical waveguides embedded a material with Kerr nonlinearities is derived and explored. The extended discrete nonlinear Schrödinger intersite nonlinearities. Attention turned towards localized solutions investigations are made from viewpoint theory breathers (DBs). Stability analysis reveals inversion stability between stationary one-site symmetric or antisymmetric two-site connected to bifurcations pair...

We review work on the Discrete Nonlinear Schrödinger (DNLS) equation over last two decades.

We discuss the properties of nonlinear localized modes in sawtooth lattices, framework a discrete Schrödinger model with general on-site nonlinearity. Analytic conditions for existence exact compact three-site solutions are obtained, and explicitly illustrated cases power-law (cubic) saturable nonlinearities. These appear as continuations linear belonging to flat dispersion band. While system mode exists only one specific ratio two different coupling constants, nonlinearity may lead...

We perform numerical investigations of the dynamical localization properties discrete nonlinear Schr\"odinger equation with periodic and deterministic aperiodic on-site potentials. The time evolution an initially single-site localized state is studied, quantities describing different aspects are calculated. find that for a large enough nonlinearity, probability finding quasiparticle at initial site will always be nonzero participation number finite all systems under study (self-trapping)....

We extend earlier work [Phys. Rev. Lett. 84, 3740 (2000)]] on the statistical mechanics of cubic one-dimensional discrete nonlinear Schrödinger (DNLS) equation to a more general class models, including higher dimensionalities and nonlinearities arbitrary degree. These extensions are physically motivated by desire describe situations with an excitation threshold for creation localized excitations, as well recent suggesting noncubic DNLS models Bose-Einstein condensates in deep optical...

We study energy propagation in locally time-periodically driven disordered nonlinear chains. For frequencies inside the band of linear Anderson modes, three different regimes are observed with increasing driver amplitude: 1) below threshold, localized quasiperiodic oscillations and no spreading; 2) time close to almost regular initially, weak chaos slow spreading for intermediate times finally strong diffusion; 3) immediate driving. The thresholds due simple bifurcations, obtained...

A one-dimensional diatomic chain with harmonic intersite potential and nonlinear external is considered (the Klein-Gordon model). Localized solutions of the corresponding differential equations frequencies inside gap linear wave spectrum--"gap breathers"--are studied numerically. The stability analysis for these performed while changing system parameters from anticontinuous to continuous limit. Two different types are considered: symmetric centered at a heavy atom antisymmetric light atom,...

We demonstrate the stability of a stationary vortex breather with vorticity S = 2 in two-dimensional discrete nonlinear Schrödinger model for square lattice and also discuss effects exciting internal sites ring. point out fundamental difficulties observing these solutions current experimental techniques. Instead, we argue that relevant initial conditions will lead to formation quasiperiodic breathers.

We investigate the resonance mechanisms for discrete breathers in finite-size Klein–Gordon lattices, when some harmonic of breather frequency enters linear phonon band. For soft on-site potentials, second-harmonic resonances typically result appearance solutions with non-zero tails, phonobreathers. However, these tails may be very weak, and small systems where frequencies are sparsely distributed, we identify 'phantom breathers' as being practically localized solutions, existing in-between...

We address the problem of directional mobility discrete solitons in two-dimensional rectangular lattices, framework a nonlinear Schr\"odinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate Peierls-Nabarro energy surfaces, which describe pseudopotential landscape for slow coherent localized excitations, corresponding continuous phase-space trajectories passing close stationary modes. Investigating two-parameter space through...