Analytic Solution for Self-regulated Collective Escape of Cosmic Rays
  from their Acceleration Sites
  Supernova remnants (SNRs), as the major contributors to the galactic cosmic
rays (CR), are believed to maintain an average CR spectrum by diffusive shock
acceleration (DSA) regardless of the way they release CRs into the interstellar
medium (ISM). However, the interaction of the CRs with nearby gas clouds
crucially depends on the release mechanism. We call into question two aspects
of a popular paradigm of the CR injection into the ISM, according to which they
passively and isotropically diffuse in the prescribed magnetic fluctuations as
test particles. First, we treat the escaping CR and the Alfven waves excited by
them on an equal footing. Second, we adopt field aligned CR escape outside the
source, where the waves become weak. An exact analytic self-similar solution
for a CR "cloud" released by a dimmed accelerator strongly deviates from the
test-particle result. The CR diffusion coefficient $D_{NL}$ is strongly
suppressed compared to its background ISM value $D_{ISM}$: $D_{NL}\sim
D_{ISM}\exp(-\Pi)<< D_{ISM}$ for sufficiently high field-line-integrated CR
partial pressure, $\Pi$. When $\Pi>>1$, the CRs drive Alfven waves efficiently
enough to build a transport barrier that strongly reduces the leakage. The
solution has a spectral break at $p=p_{br}$, where $p_{br}$ satisfies the
following equation $D_{NL}(p_{br})\simeq z^{2}/t$.
