Multicomponent nonlinear dynamical systems inspired by the Toda lattice

A matrix nonlinear model on an infinite lattice is proposed. The model admits the Lax representation as being the natural matrix generalization of the ordinary Toda lattice. As a result the system is exactly integrable and possesses the Baecklund auto-transformation useful in generating nontrivial solutions. The auxiliary matrix integrable nonlinear system reducible to the Baecklund auto-transformation of interest is developed. The simplest regular and nonregular solutions of the basic model are found and analysed. Thus, in parallel with the kink-type field component the regular solution was observed to consist of field components evolving as solitons on finite backgrounds. In a particular case of four-component realization, the model permits both Lagrangian and Hamiltonian formulations, while the corresponding Baecklund transformation is proved to be the canonical one. We believe that similar statements can be made for any multi-component version of the general matrix model, provided an appropriate parametrization of field variables is adopted. (author)