Effect of lattice ribbonization via the background-controlled inter-site resonant interactions in nonlinear integrable systems

Abstract

Any nonlinear integrable dynamical system of coherently coupled intra-site excitations associated with one or another semi-discrete zero curvature representation, whose auxiliary spectral matrix has the determinant dependent on the spectral parameter, is claimed to possess the set of concomitant fields dependent on the set of basic field variables. The nonzero background level of concomitant fields turns out to be the source of additional background-controlled inter-site coherent (resonant) interactions between the basic excitations. As a result, the purely one-dimensional primary spatial lattice dealing with the auxiliary linear problem is converted into a ladder-like arrangement of sites in the secondary lattice ribbon serving as a spatial support for the basic fields. The effect of lattice ribbonization gives rise to the rather nonstandard Poisson structure involved into the Hamiltonian equations as well as to the essential enrichment of the whole system’s dynamics. For example, depending on the background level of concomitant fields a system is able to experience the crossover in the very origin of its excitations. Nevertheless, a system is found to afford the soliton and multi-soliton solutions with the soliton parameters accumulating the contributions of both the usual inter-site resonant couplings and the additional background-controlled ones. Despite of its nontriviality the challenging problem of standardization is proved to be successfully solved in terms of properly chosen physically corrected fields at least for the certain particular nonlinear integrable system on a lattice with two structural elements in the unit cell.