  We explore breather propagation in the damped oscillatory chain with
essentially nonlinear (non-linearizable) nearest-neighbour coupling.
Combination of the damping and the substantially nonlinear coupling leads to
rather unusual two-stage pattern of the breather propagation. The first stage
occurs at finite fragment of the chain and is characterized by power-law decay
of the breather amplitude. The second stage is characterized by extremely small
breather amplitudes that decay hyper-exponentially with the site number. Thus,
practically, one can speak about finite penetration depth of the breather. This
phenomenon is referred to as breather arrest (BA). As particular example, we
explore the chain with Hertzian contacts. Dependencies of the breather
penetration depth on the initial excitation and on the damping coefficient on
the breather penetration depth obey power laws. The results are rationalized by
considering beating responses in a system of two damped linear oscillators with
strongly nonlinear (non-linearizable) coupling. Initial excitation of one of
these oscillators leads to strictly finite number of beating cycles. Then, the
beating cycle in this simplified system is associated with the passage of the
discrete breather between the neighbouring sites in the chain. Somewhat
surprisingly, this simplified model reliably predicts main quantitative
features of the breather arrest in the chain, including the exponents in
numerically observed power laws.
