Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 089, 17 pages      arXiv:2509.17976      https://doi.org/10.3842/SIGMA.2025.089

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

Oleksiy O. Vakhnenko a and Vyacheslav O. Vakhnenko b
a) Bogolyubov Institute for Theoretical Physics of NAS of Ukraine, 14-B Metrologichna Str., Kyïv 03143, Ukraine
b) Subbotin Institute of Geophysics of NAS of Ukraine, 63-B Bohdan Khmel'nyts'kyy Str., Kyïv 01054, Ukraine

Received May 24, 2025, in final form September 17, 2025; Published online October 22, 2025

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.

Key words: Lax integrability; quasi-one-dimensional lattice; multicomponent system; nonlinear dynamics; $\mathcal{PT}$ symmetry.

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