  In the present work, the Yang-Mills (YM) quantum-wave excitations of the
classical homogeneous YM condensate have been studied in quasi-classical
approximation. The formalism is initially formulated in the Hamilton gauge and
is based upon canonical quantisation in the Heisenberg representation. This
canonical framework is then extended and related to YM dynamics in arbitrary
gauge and symmetry group containing at least one $SU(2)$ subgroup. Such generic
properties of the interacting YM system as excitation of longitudinal wave
modes and energy balance between the evolving YM condensate and waves have been
established. In order to prove these findings, the canonical quasi-classical YM
system "waves + condensate" in the pure simplest $SU(2)$ gauge theory has been
thoroughly analysed numerically in the linear and next-to-linear approximations
in the limit of small wave amplitudes. The effective gluon mass dynamically
generated by wave self-interactions in the gluon plasma has been derived. A
complete set of equations of motion for the YM "condensate + waves" system
accounting for second- and third-order interactions between the waves has been
obtained. In the next-to-linear approximation in waves we have found that due
to interactions between the YM waves and the YM condensate, the latter looses
its energy leading to the growth of amplitudes of the YM wave modes. A similar
effect has been found in the maximally-supersymmetric ${\cal N}=4$ Yang-Mills
theory as well as in two-condensate $SU(4)$ model. Possible implications of
these findings to Cosmology and gluon plasma physics have been discussed.
