arXiv · cond-mat/0402072
On classification of intrinsic localized modes for the Discrete Nonlinear Schrödinger Equation
Abstract
We consider localized modes (discrete breathers) of the discrete nonlinear Schrödinger equation $i\frac{dψ_n}{dt}=ψ_{n+1}+ψ_{n-1}-2ψ_n+σ|ψ_n|^2ψ_n$, $σ=\pm1$, $n\in \mathbb{Z}$. We study the diversity of the steady-state solutions of the form $ψ_n(t)=e^{iωt}v_n$ 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 $ω<ω^*\approx -3.4533$ 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.
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G. L. Alfimov, V. A. Brazhnyi, V. V. Konotop. 2004-02-03. On classification of intrinsic localized modes for the Discrete Nonlinear Schrödinger Equation. https://doi.org/10.1016/j.physd.2004.02.001
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