Physical insight into the semi-discrete nonlinear integrable systems with the true and false multicomponentness

The critical overview of controversial, curious and fundamental aspects of classical semi-discrete nonlinear integrable systems is presented from the standpoint of their physical applicability and physically meaningful formulation. The paper gives a number of practical suggestions how to reformulate certain two-component and multicomponent semi-discrete nonlinear integrable systems of nonlinear Schrödinger type in true physical terms familiar for physicists, as well as how to enrich the multicomponent nonlinear Schrödinger systems by the interchain linear interactions important for the realistic modeling of regular quasi-one-dimensional physical objects. In particular, the physically adequate canonization of the four-component nonlinear integrable system with the background-controlled couplings was shown to reveal the critical crossover between the bright–bright and bright–dark regimes of excitation dynamics. We also pay attention to the nearly integrable nonlinear Schrödinger system on a multileg ladder lattice that admits the strict canonical Hamiltonian formulation in terms of physically meaningful field variables, as well as on its completely integrable spatially continuous analog. On the other hand, we disclose the false multicomponent treatment of systems with the so-called branched dispersion. We believe that our experience will be helpful for the researchers dealing with and developing the nonlinear integrable dynamical systems on quasi-one dimensional lattices potentially applicable in different branches of physical science.