Nonlinearity 10 (1997) 1151–1178. Printed in the UK PII: S0951-7715(97)81838-5 Existence and stability of quasiperiodic breathers in the discrete nonlinear Schr¨ odinger equation Magnus Johansson† and Serge Aubry‡ † Department of Mathematical Modelling, The Technical University of Denmark, DK-2800 Lyngby, Denmark ‡ Laboratoire L´ eon Brillouin (CEA-CNRS), CE Saclay, 91191 Gif-Sur-Yvette Cedex, France Received 10 February 1997, in final form 11 June 1997 Recommended by A Kupiainen Abstract. We show that the discrete nonlinear Schr¨ odinger (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. PACS numbers: 0320, 6320P, 6320R, 7138 1. Introduction Recently, there has been a rapidly increasing interest in the investigation of localized modes in nonlinear discrete systems. In particular, the existence of exact solutions which are periodic in time and exponentially localized in space was rigorously proven under rather general conditions for a large class of nonlinear lattice equations in the regime of large nonlinearity [1]. In analogy with the breather solutions to the continuous sine-Gordon equation, which have similar properties, these solutions were termed discrete breathers. One particular example where discrete breathers were proved to exist is the discrete nonlinear Schr¨ odinger (DNLS) equation, whose one-dimensional version reads: i ˙ ψ n + C(ψ n+1 + ψ n−1 ) +|ψ n | 2 ψ n = 0. (1) The DNLS equation, which is non-integrable, has been used to model the self-localization (or self-trapping) of energy in various physical disciplines, such as polaron formation in 0951-7715/97/051151+28$19.50 c  1997 IOP Publishing Ltd and LMS Publishing Ltd 1151