Semidiscrete Integrable Nonlinear Schrӧdinger System with Background-Controlled Intersite Resonant Coupling. Short Summary of Key Properties

Abstract

The most featured items characterizing the semidiscrete nonlinear Schr¨odinger system with background-controlled intersite resonant coupling are summarized. The system is shown to be integrable in the Lax sense that makes it possible to obtain its soliton solutions in the framework of a properly parametrized dressing procedure based on the Darboux transformation accompanied by the implicit form of B¨acklund transformation. In addition, the system integrability inspires an infinite hierarchy of local conservation laws, some of which were found explicitly in the framework of the generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem, and its dynamics is embedded into the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of the intersite resonant coupling. As a consequence, it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. At tuning the main background parameter, we are able to switch system’s dynamics between two essentially different regimes separated by the critical point. The physical implications of system’s criticality become evident after a rather sophisticated procedure of canonization of basic field variables. There are two variants to standardize the system equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus, in the under-critical region, the system supports the bright excitations in both subsystems; while, in the over-critical region, one of the subsystems converts into the subsystem of dark excitations.