Nonlinear localization in modulated and incommensurate (self-modulated) optical structures

The summary form only given. Both theoretical and experimental investigations of structure and dynamics of 2D mono-atomic surface layers, layers of adatoms and ID adatoms chains on the surface are of great interest now. Usually these adatoms form the incommensurate structures on the surface. In the simplest approach this problem can be studied in the framework of sin-Gordon model. (The substrate is considered to be rigid.) In this model an incommensurate structure represents the periodic array of topological solitons (kinks). The linear excitations in this structure (Swihart mode) are well-known in the theory of Josephson junction. The additional kink propagating through such an incommensurate structure represents the elementary nonlinear excitation in the system. The exact analytical solution for this dynamical soliton was obtained-by using Darboux transformation for SGE. It is interesting that there exists the minimal nonzero velocity of such a kink. That velocity coincides with Swihart velocity. The knowledge of this soliton solution allows us to obtain through the Baklund transformation the more complicated exact solution for envelope two-parametric soliton. In particular limit case this solution describes the small amplitude nonlinear excitations in the incommensurate surface structure. On the other hand, it represents the so-called "gap" soliton with the frequencies in the gap above Swihart mode. Also we propose the qualitative approach to nonlinear dynamics of the incommensurate structure in terms of a chain of quasiparticles with definite nonlinear interaction. (The kinks of incommensurate structure play the role of these quasiparticles.) In long-wave small-amplitude approximation Boussinesq equation is obtained for this effective chain. The well-known soliton solutions of Boussinesq equation are consistent with obtained exact solutions in the small-amplitude limit.