Code from Nonlinear system of ????????-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

The nonlinear dynamics of coupled 𝒫𝒯-symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.