  In this paper we present the first formulation of the theory of non-linear
particle acceleration in collisionless shocks in the presence of neutral
hydrogen in the acceleration region. The dynamical reaction of the accelerated
particles, the magnetic field amplification and the magnetic dynamical effects
on the shock are also included. The main new aspect consists however in
accounting for charge exchange and ionization of neutral hydrogen, which
profoundly change the structure of the shock, as discussed in our previous
work. This important dynamical effect of neutrals is mainly associated to the
so-called neutral return flux, namely the return of hot neutrals from the
downstream region to the upstream, where they deposit energy and momentum
through charge exchange and ionization. We also present the self-consistent
calculation of Balmer line emission from the shock region and discuss how to
use measurements of the anomalous width of the different components of the
Balmer line to infer the cosmic ray acceleration efficiency in supernova
remnants showing Balmer emission: the broad Balmer line, which is due to charge
exchange of hydrogen atoms with hot ions downstream of the shock, is shown to
become narrower as a result of the energy drainage into cosmic rays, while the
narrow Balmer line, due to charge exchange in the cosmic-ray--induced
precursor, is shown to become broader. In addition to these two well-known
components, the neutral return flux leads to the formation of a third component
with intermediate width: this too contains information on ongoing processes at
the shock.
