Fundamentals of Non-relativistic Collisionless Shock Physics: II. Basic   Equations and Models
  This paper develops the basic sets of equations which lead to the conservation laws describing collisionless plasma shock waves. We discuss the evolution of shock waves by wave steepening, derive the Rankine-Hugoniot conditions for magnetogasdynamic shocks, discuss various analytical models of shock formation, and discuss the basic instabilities which may become important in collisionless shock physics. We then present a survey of the theory of anomalous resistivity in the quasilinear limit and beyond and discuss mechanisms of shock particle reflection as far as they have been investigated in the published literature. The content of the chapter is the following: 1. Wave steepening, describing simple waves and steepening due to nonlinearity, balnced by dissipation in Burgers' shocks, by dispersive effects in the Korteweg-de Vries equation, the Sagdeev-Potential method, 2. Basic equations, presenting kinetic theory and the transition to moment equations in the fluid description, 3. Rankine-Hugoniot relations, giving the jump conditions across shocks, explicit MHD solution for perpendicular shocks and parallel shocks, and high Mach number conditions, 4. Waves and instabilities, giving the general dispersion relation, describing low-$\beta$-shocks, whistler and Alfv\'en shocks, the various shock-relevant instabilities, 5. Anomalous transport for the various electrostatic wave-particle interactions, general description of anomalous resistivity, shock particle reflection from potential and specularly, hole formation, 6. Briefing on numerical simulation techniques, giving a short idea on this important field and its methods. 
R. A. Treumann and C. H. Jaroschek