Analysis of stability of solitons in one-dimensional lattices

We develop a direct analysis of the soliton stability problem for the simplest model of a closed dynamical lattice with the potential of the nearest-neighbor interaction containing quadratic and quartic terms. In the lowest approximation the soliton is represented as a discrete step function, its height being an arbitrary parameter. In this approximation, the stability problem is solved analytically. The soliton proves to be always stable; a single localized eigenmode of small disturbances is found, all other eigenmodes being delocalized. In the next approximation, the soliton is taken as a combination of two steps, so that it has an inner degree of freedom. Using numerical methods, we demonstrate that in this approximation the soliton remains stable; a second localized eigenmode is found in a certain parametric region.