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

We consider localized modes (discrete breathers) of the discrete nonlinear Schrödinger equation i(dψn/dt)=ψn+1+ψn−1−2ψn+σ|ψn|2ψn, σ=±1, n∈Z. We study the diversity of the steady-state solutions of the form ψn(t)=eiωtvn and the intervals of the frequency, ω, of their existence. The base for the analysis is provided by the anticontinuous limit (ω 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 ω<ω∗≈−3.4533 and the point ω∗ is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for ω>ω∗ and give the complete table of them for the solutions with codes consisting of less than four symbols.