Distinctive Features of the Integrable Nonlinear Schr¨odinger System on a Ribbon of Triangular Lattice

Abstract

The dynamics of an integrable nonlinear Schr¨odinger system on a triangular-lattice ribbon is shown to be critical against the value of background parameter regulated by the limiting values of concomitant fields. Namely at the critical point, the number of basic field variables is reduced by half and the Poisson structure of the system becomes degenerate. On the other hand, outside the critical point, the form of Poisson structure turns out to be an essentially nonstandard one, and the meaningful procedure of its standardization leads inevitably to the breaking of the mutual symmetry between the standardized basic subsystems. There are two possible realizations of such an asymmetric standardization, each giving rise to a total suppression of field amplitudes in one of the standardized basic subsystems at the critical value of background parameter. In the undercritical region the standardized basic field amplitudes acquire the meaning of probability amplitudes of some nonequivalent intracell bright excitations, whereas in the overcritical region such an interpretation is proven to be incorrect. A proper analysis shows that the overcritical region could be thought as the region of coexistence between the standardized subsystems of bright and dark excitations.