Self-consistent interaction of neutrals and shocks in the local   interstellar medium
  The problem of hydrogen neutrals interacting with the heliospheric bow shock region has received considerable attention recently, motivated primarily by the hope that the Voyager spacecraft may soon encounter the first of the heliospheric boundaries. The complexity of the charge exchange interactions has limited our analytic understanding so far. In this work we develop a semi-analytic one-dimensional model based upon a double expansion technique to investigate the self-consistent interaction of shock with neutrals. The underlying method uses the Boltzmann transport equation that describes neutral transport, and is coupled to plasma protons predominantly through charge exchange processes. The gyrotropic distribution function of the neutrals is first expanded in terms of Legendre polynomials followed by associated Laguerre harmonics. The resulting set of equations can be cast into a square matrix whose eigenvalues depend upon sources and the plasma distribution. The interstellar protons, in a high $\beta$ limit, are described by a hydrodynamic fluid with source terms that couple neutrals and plasma and further modify both distributions in a self-consistent manner. 
Dastgeer Shaikh, G. P. Zank and N. Pogorelov