47 Lenin Ave.,

61103 Kharkiv,

UKRAINE

and

4a Chaikovsky Str.,

61002 Kharkiv,

UKRAINE

E-mail: bogdan@ilt.kharkov.ua

**Solitons in the finite-size modulated systems described
by the discrete nonlinear Schr\"odinger-type equations**

**Abstract:**

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.