Evolution of coupled classical fields

We study the evolution of the coupled scalar and fermion fields within the classical field theory. We examine the case of N coupled fields in (1+3)-dimensional space. The general expressions for the fields distributions are obtained. The particular case of two fields in (1+1)-dimensional space is carefully studied. We obtain the expressions for the averaged fields intensities and show that in the relativistic limit they are similar to the usual transition probabilities formulae of neutrino oscillations.