Peculiarities of dynamics of one-dimensional discrete systems with interaction extending beyond nearest neighbors, and the role of higher dispersion in soliton dynamics

In the analysis of dynamics of an ideal system as well as a system with poi nt defects, the role of interaction is considered not only for the nearest neighbors. The Green's function is constructed for steady-state vibrations of a chain at all possible frequencies. It is shown that, if the interactio n with the next-to-nearest neighbors is taken into account, the Green's fun ction inevitably becomes double partial, the nature of its two components d epending significantly on its eigenfrequency. It is found that the Green's function for frequencies of the continuous spectrum of small vibrations has one component of the plane wave type, while the other component is localiz ed near the source of perturbations. Such a Green's function describes the so-called quasilocal vibrations. At certain discrete frequencies falling in the continuous spectrum, the quasilocal vibration is transformed into loca l vibration (that does not propagate to infinity). The conditions of applic ability of differential equations with fourth spatial derivative are analyz ed for describing the longwave vibrations of the atomic chain. Relations be tween parameters of atomic interactions permitting the use of such equation s are formulated. Asymptotic forms of soliton fields in a nonlinear medium with spatial dispersion are discussed. It is shown that most of the soliton parameters are determined by the dispersion relation for the linearized eq uation. (C) 1999 American Institute of Physics. [S1063-777X(99)01407-3].