On classification of intrinsic localized modes for the Discrete Nonlinear Schr\"{o}dinger Equation

We consider localized modes (discrete breathers) of the discrete nonlinear Schr\"{o}dinger equation $i\frac{d\psi_n}{dt}=\psi_{n+1}+\psi_{n-1}-2\psi_n+\sigma|\psi_n|^2\psi_n$, $\sigma=\pm1$, $n\in \mathbb{Z}$. We study the diversity of the steady-state solutions of the form $\psi_n(t)=e^{i\omega t}v_n$ 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 $\omega<\omega^*\approx -3.4533$ 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.