Oscillons and quasibreathers in the 4 Klein-Gordon model

Strong numerical evidence is presented for the existence of a continuous family of time-periodic solutions with ``weak'' spatial localization of the spherically symmetric nonlinear Klein-Gordon equation in $3+1$ dimensions. These solutions are ``weakly'' localized in space in that they have slowly decaying oscillatory tails and can be interpreted as localized standing waves (quasibreathers). By a detailed analysis of long-lived metastable states (oscillons) formed during the time evolution, it is demonstrated that the oscillon states can be quantitatively described by the weakly localized quasibreathers. It is found that the quasibreathers and their oscillon counterparts exist for a whole continuum of frequencies.