Dromions bound states

The asymptotic perturbation (AP) method is applied to the study of the nonlinear Klein–Gordon equation in 3+1 dimensions with harmonic potential and external periodic excitation supposed to be in primary resonance with the frequency of a generic mode. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing of the harmonic terms with a simple iteration. Standard quantum mechanics can be used to derive the lowest order approximate solution and amplitude and phase modulation equations are obtained. External force–response and frequency–response curves are found and the existence of dromions trapped in bound states is demonstrated.