Integrable Twelve-Component Nonlinear Dynamical System on a Quasi-One-Dimensional Lattice

TL;DR

The authors address modeling transport phenomena on quasi-one-dimensional lattices by constructing two new multicomponent semi-discrete integrable systems within the semi-discrete AKNS framework. They derive a twelve-component and a six-component model, obtain their primary prototype equations, and implement two reductions to yield coupled multi-chain dynamics with external drive and magnetic-field-like couplings. Central contributions include the systematic use of local conservation laws to fix sampling functions, the identification of admissible symmetries (complex conjugation and space-time reversal), and the explicit structure of the resulting twelve- and six-component systems, including their current-density interpretations. The work lays a foundation for future analytic solutions and Hamiltonian formulations, with potential applications to transport in long quasi-one-dimensional systems and macromolecular structures under external fields.

Abstract

Bearing in mind the potential physical applicability of multicomponent completely integrable nonlinear dynamical models on quasi-one-dimensional lattices we have developed the novel twelve-component and six-component semi-discrete nonlinear inregrable systems in the framework of semi-discrete Ablowitz-Kaup-Newell-Segur scheme. The set of lowest local conservation laws found by the generalized direct recurrent technique was shown to be indispensable constructive tool in the reduction procedure from the prototype to actual field variables. Two types of admissible symmetries for the twelve-component system and one type of symmetry for the six-component system have been established. The mathematical structure of total local current was shown to support the charge transportation only by four of six subsystems incorporated into the twelve-component system under study. The twelve-component system is able to model the actions of external parametric drive and external uniform magnetic field via time dependencies and phase factors of coupling parameters.