On the asymptotic stability of monotonic electrostatic collisionless shock waves

Recently the question of existence and stability of laminar one-dimensional coffisionless shock waves has received some new attention (Montgomery & Joyce 1969). These electrostatic shock waves are usually thought of as being generated by ion acoustic waves (pulses) in a plasma with T_e T_i Apart from the well- known class of shock solution with oscillatory structure (Moiseev & Sagdeev 1963) there exist, at least within the framework of Vlasov theory, monotonic transitions from one asymptotic state to another (Montgomery & Joyce 1969). For these shock-wave like solutions it is easy to see that the electrons cannot be in thermal equilibrium (have a Maxwellian distribution) in the downstream state, if they are so in the upstream state. Hence it appears to be natural to investigate first the stability of the homogeneous downstream state (which tells us if the transition is possible at all) by investigating the properties of the class of permitted distribution functions for the electrons, before attacking the much more involved problem of the possible instabilities arising from the inhomogeneitythe transition region of the shock.