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 $i\frac{d{\psi }_n}{dt}={\psi }_{n+1}+{\psi }_{n-1}-2{\psi }_n+\sigma \mathrm{\mid }{\psi }_n{\mathrm{\mid }}^2{\psi }_n$, $\sigma =\pm 1$, $n\in \mathbb{Z}$. 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 $\omega <{\omega }^{\ast }\approx -3.4533$ and the point ${\omega }^{\ast }$ is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for $\omega >{\omega }^{\ast }$ and give the complete table of them for the solutions with codes consisting of less than four symbols.