  For the James breathers in the K2-K3-K4 chain and for breathers in the
K4-chain, we prove numerically that these dynamical objects are not strictly
time-periodic. Indeed, for the both cases, there exist certain deviations in
the vibrational frequencies of the individual particles, which certainly exceed
the possible numerical errors. We refer to the dynamical objects with such
properties as quasibreathers. For the K4-chain, a rigorous investigation of
existence and stability of the breathers and quasibreathers is presented. In
particular, it is proved that they are stable up to a certain strength of the
intersite part of the potential with respect to its on-site part. We conjecture
that the main results of this paper are also valid in the general case and,
therefore, it seems that one must speak about quasibreathers rather than about
strictly time-periodic breathers.
