Coupling-governed metamorphoses of the integrable nonlinear Schrödinger system on a triangular-lattice ribbon

• The integrable nonlinear Schrödinger system on a triangular-lattice ribbon is inclined to metamorphoses.
• The system under study is capable to incorporate the effect of external linear potential.
• The system criticality against the background parameter reduces the number of independent field variables.
• At critical point the system Poisson structure becomes degenerate.
• The effect of criticality is elucidated by the system one-soliton solution.

The variativity of governing coupling parameters in the integrable nonlinear Schrödinger system on a triangular-lattice ribbon is shown to ensure the important qualitative rearrangements in the system dynamics. There are at least the two types of system crucial modifications stipulated by the two types of governing parameters. Thus the longitudinal coupling parameters regulated mainly by the background values of concomitant field variables are responsible for the bifurcation of primary integrable nonlinear system into the integrable nonlinear system of Ablowitz–Ladik type. As a consequence in a critical point the number of independent field variables is reduced by a half and the system Poisson structure turns out to be degenerate. On the other hand the transverse coupling parameters regulated basically by the choice of their a priori arbitrary dependencies on time are capable to incorporate the effect of external linear potential. As a consequence the primary integrable nonlinear system with appropriately adjusted parametrical driving becomes isomorphic to the system modeling the Bloch oscillations of charged nonlinear carriers in an electrically biased ribbon of triangular lattice. The multi-component structure of basic integrable system alongside with the attractive character of system nonlinearities has predetermined the logic of whole consideration.