Nonlinear Schrödinger Equation with a Short-Range Compensating Field

Within the self-consistent Maxwell-Pauli theory, a nonlinear Schrödinger equation with a short-range compensating field is derived. Stationary and non-stationary solutions of the obtained nonlinear Schrödinger equation for the hydrogen atom are investigated. It is shown that spontaneous emission and the associated rearrangement of the internal structure of the atom, which is traditionally called a spontaneous transition, have a simple and natural description within the classical field theory without any quantization and additional hypotheses. The solution of the nonlinear Schrödinger equation showed that, depending on the frequency of spontaneous emission, the compensating field behaves differently. At relatively low frequencies of spontaneous emission, there is no radiation (waves) of the short-range compensating field and this field does not carry away energy. In this case, the damping rate of the spontaneous emission coincides with that obtained in quantum electrodynamics (QED). At relatively high frequencies of spontaneous emission, radiation (waves) of the compensating field arises, which, along with electromagnetic radiation, carry away some of the energy. In this case, the damping rate of the spontaneous emission is greater (up 1.5 times) than that predicted by QED.