Four-component integrable systems inspired by the Toda and the Davydov–Kyslukha models

Two types of general nonlinear integrable systems on infinite quasi-one-dimensional regular lattices are proposed. In accordance with the Mikhailov reduction group theory both general systems turn out to be underdetermined, thereby permitting numerous reduced systems written in terms of true field variables. Each reduced system thus obtained can be regarded as an integrable version of two particular coupled subsystems, and under the appropriate choice of coupling parameters any reduced systems can manifest the symmetry under the space and time reversal (PT-symmetry). Thus, we have managed to unify the Toda-like vibration subsystem and the self-trapping-like exciton subsystem into a single integrable system, thereby substantially extending the range of realistic physical problems that can be rigorously modeled. Several lowest conserved densities associated with either of the relevant infinite hierarchies of local conservation laws are found explicitly in terms of prototype field functions. •Semi-discrete integrable nonlinear systems incorporating two distinct subsystems are proposed.•The lowest conserved densities for each integrable nonlinear system are found explicitly.•The two-subsystem dynamics provides the modeling mechanism for transport phenomena.•Challenging problems related to the proposed coupled integrable systems are formulated.