2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/93461Hamiltonians of a wide-spread class of strongly coupled quantum system models are expressed as nonlinear functions of $sl(2)$ generators. It enables us to use the $sl(2)$ formalism, in particular, $sl(2)$ generalized coherent states (GCS) for solving both spectral and evolution tasks. In such a manner, using standard variational schemes with $sl(2)$ GCS as trial functions we find new analytical expressions for energy spectra and non-linear evolution equations for cluster dynamics variables in mean-field approximations which are beyond quasi-harmonic ones obtained earlier. General results are illustrated on certain concrete models of quantum optics and laser physics.19 pages, LATEXQuantum Physicstext