Quasibreathers as a generalization of the concept of discrete breathers.
The majority of dynamical objects, which demonstrate energy localization in nonlinear lattices, represent quasibreathers rather than strictly time-periodic discrete breathers since, as a rule, there exist certain deviations in vibrational frequencies of the individual particles exceeding the possible numerical errors. We illustrate this idea with the James breathers in the K2-K3-K4 chain and with quasibreathers in the K4 chain. For the latter case, 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 quasibreathers play a fundamental role in the problem of energy localization in more realistic nonlinear lattices, as well. The difference between breathers and quasibreathers can be characterized by the mean square deviation of the frequencies of individual particles.