On the Nonlinear Two- and Three-Dimensional Klein–Gordon Equations Allowing Localized Solutions with Beatings of Coupled Oscillators

Equations for two and three scalar fields, which allow localized solutions with beatings of coupled oscillators, have been presented. The amplitude of oscillations of a localized perturbation for one field decreases periodically gradually to a minimum and the amplitudes of the other scalar fields increase to a maximum; then, the reverse process occurs. In this case, all fields except for one are initially either in the state of a background solution with a small amplitude or equal to zero. Such solutions can be interesting due to analogy with neutrino oscillations. Equations of motion, where the perturbation of one of the components is obligatorily accompanied by the perturbation of the second and third components even in zeroth background state, have also been presented. It has been shown that these equations satisfy the energy conservation law.