On classification of intrinsic localized modes for the discrete nonlinear Schrodinger equation

We consider localized modes (discrete breathers) of the discrete nonlinear Schrodinger equation i(dpsi(n)/dt) = psi(n+1) + psi(n-1) - 2psi(n) + sigma\psi(n)\(2) psi(n), sigma = +/-1, n epsilon Z. We study the diversity of the steady-state solutions of the form psi(n) (t) = e(iomegat) upsilon(n), and the intervals of the frequency, omega, of their existence. The base for the analysis is provided by the anticontinuous limit (omega negative and large enough) where all the solutions can be coded by the sequences of three symbols "-", "0" and Using dynamical systems approach we show that this coding is valid for omega < omega* approximate to -3.4533 and the point omega* is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for omega > omega* and give the complete table of them for the solutions with codes consisting of less than four symbols. (C) 2004 Elsevier B.V. All rights reserved.