Solitons in the finite-size modulated systems described by the discrete nonlinear Schr\"odinger-type equations
The contribution is devoted to the investigation of soliton analogues and their stability in the modulated chain of anharmonic oscillators of a finite size. Dynamics of the system obeys the discrete nonlinear Schr\"odinger-type equations. Two kinds of modulated systems have been studied: (i) one type is the generalization of the usual discrete nonlinear Schr\"odinger (DNLS) model for the case of alternation of normal frequency parameters. The second is the analogous generalization of the exactly integrable Ablowitz-Ladik system. In the case of the modulated DNSL model the process of appearance of analogue of the gap soliton in the systems of four, eight and twelve particles have been investigated. Nonlinear monochromatic oscillations of these systems have been studied analytically and numerically, and dependences of frequencies of the oscillations on the integral of the number of states are calculated. The problem of monochromatic oscillations of four oscillators has been reduced to four independent ones, namely the problems of stationary states of two different coupled nonlinear oscillators. The stability problem of the solitonic solutions has been formulated and solved. It is shown that the gap soliton conserves its stability after transformation into the out-of-gap soliton solution. In the modulated Ablowitz-Ladik system explicit solutions describing principal nonlinear oscillations and quasi-classical spectra have been found analytically. Comparison of spectra of soliton analogues in the finite-size chain and the quasi-classical soliton spectrum in the infinite integrable Ablowitz-Ladik system has been carried on.