Breathers in nonlinear lattices: existence, linear stability and quantization.

*(English)* Zbl 1194.34059
Summary: 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.

##### MSC:

34C15 | Nonlinear oscillations, coupled oscillators (ODE) |

35Qxx | PDE of mathematical physics and other areas |

65L10 | Boundary value problems for ODE (numerical methods) |

70K99 | Nonlinear dynamics (general mechanics) |

82C20 | Dynamic lattice systems and systems on graphs |