  The time evolution of magnons subject to a time-dependent microwave field is
usually described within the so-called "S-theory", where kinetic equations for
the distribution function are obtained within the time-dependent Hartree-Fock
approximation. To explain the recent observation of "Bose-Einstein condensation
of magnons" in an external microwave field [Demokritov et al., Nature 443, 430
(2006)], we extend the "S-theory" to include the Gross-Pitaevskii equation for
the time-dependent expectation values of the magnon creation and annihilation
operators. We explicitly solve the resulting coupled equations within a simple
approximation where only a single condensed mode is retained. We also
re-examine the usual derivation of an effective boson model from a realistic
spin model for yttrium-iron garnet films and argue that in the parallel pumping
geometry (where both the static and the time-dependent magnetic field are
parallel to the macroscopic magnetization) the time-dependent Zeemann energy
cannot give rise to magnon condensation.
