Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation

Abstract

We show that the discrete nonlinear Schrödinger (DNLS) equation exhibits exact solutions which are quasiperiodic in time and localized in space if the ratio between the nonlinearity and the linear hopping constant is large enough. These quasiperiodic breather solutions, which also exist for a generalized DNLS equation with on-site nonlinearities of arbitrary positive power, can be constructed by continuation from the anticontinuous limit (i.e. the limit of zero hopping) of solutions where two (or more) sites are oscillating with two incommensurate frequencies. By numerical continuation from the anticontinuous limit, some quasiperiodic breathers are explicitly calculated, and their domain of existence is determined. Using Floquet analysis, we also show that the simplest quasiperiodic breathers are linearly stable close to the anticontinuous limit, and we determine numerically the stability boundaries. The nature of the bifurcations occurring at the boundaries of the stability and existence regions, respectively, is investigated by analysing the band structure of the corresponding Newton operator. We find that the way in which the breather stability and existence is lost depends qualitatively on the ratio between its frequencies. In some cases the two-site breather becomes unstable with respect to a pinning mode, so that applying a small perturbation results in a splitting of the breather into one pinned and one moving part. In other cases, the breather develops an extended tail as some harmonic of its frequencies enters the linear phonon band and becomes a `phonobreather', which was found to be linearly stable in some domain of parameters.