Breathers In Nonlinear Lattices Research Articles

Unidirectional transport of wave packets through tilted discrete breathers in nonlinear lattices with asymmetric defects.

We consider the transfer of lattice wave packets through a tilted discrete breather (TDB) in opposite directions in the discrete nonlinear Schrödinger model with asymmetric defects, which may be realized as a Bose-Einstein condensate trapped in a deep optical lattice, or as optical beams in a waveguide array. A unidirectional transport mode is found, in which the incident wave packets, whose energy belongs to a certain interval between full reflection and full passage regions, pass the TDB only in one direction, while in the absence of the TDB, the same lattice admits bidirectional propagation. The operation of this mode is accurately explained by an analytical consideration of the respective energy barriers. The results suggest that the TDB may emulate the unidirectional propagation of atomic and optical beams in various settings. In the case of the passage of the incident wave packet, the scattering TDB typically shifts by one lattice unit in the direction from which the wave packet arrives, which is an example of the tractor-beam effect, provided by the same system, in addition to the rectification of incident waves.

Breathers in nonlinear lattices: Existence, linear stability and quantization

It has been proved by the principle of anti-integrable limit (renamed here anti-continuous limit) and the implicit function theorem that, under rather weak hypotheses, there exist spatially localized and time periodic solutions (“breathers”) in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong. The models can be translationally invariant or not and be of any dimension. There are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions (“multibreathers”). The whole set of these solutions can be labeled by a discrete coding sequence. The condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called “phonobreather” which correspond to special codes (which can be also spatially chaotic) and where no oscillator is at rest. These results extend almost without change to models with coupled rotors. The existence of “rotobreathers” which are exact solutions consisting of a single rotor rotating while the other rotors oscillate, is proved. There are also multirotobreather solutions and “phono-rotobreathers”, etc. It is also shown that the phase of the single breathers constituting a multibreather (or multirotobreather) solution can be submitted to a “phase torsion”. These new dynamical structures carry a nonzero energy flow. In models with two dimensions and more, the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow. The dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti-continuous limit. The existence of two kinds of “polarobreathers”, which are spatially exponentially localized dynamical solutions, is proved for a small enough electronic transfer integral. The polarobreathers of the first kind include the standard static polaron. The lattice configuration is static (time independent) when the electron is in an excited state (i.e. the electronic oscillator oscillates periodically). They can be spatially chaotic. A polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation. The most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site. The standard Floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the Newton operator involved in the implicit function theorem. The Krein theory of bifurcations is reinterpreted from this point of view. The linear stability of the single breather (and rotobreather) solutions is then proved at least at small enough coupling. Finally, it is shown that in the quantum version of translationally invariant classical models (where there cannot exist in principle any strictly localized excitation) there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers. The bandwidth of these excitations goes exponentially to zero with the number of bound phonons. This theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist. These numerical applications are currently written elsewhere.

Slow Relaxation Phenomena Induced by Breathers in Nonlinear Lattices

We study relaxation properties of one dimensional nonlinear lattices which are initially thermalized and subsequently put intp contact with a cold bath simulated by absorbing boundary conditions. We observe a nonexponential lattice energy relaxation in contrast to the standard exponential relaxation law of the corresponding linear system. We connect the long-tail relaxation behavior with the presence of long-lived nonlinear localized modes. The mobility of the breathers is shown to play a substantial role in the lattice relaxation properties.

Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit

Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi - Pasta - Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate `rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.