Precursor of ion-acoustic quasishock wave in collisionless plasma

A weak, ion-acoustic quasishock wave which propagates into a uniform state is considered. A relevant initial value problem of the Vlasov equation is solved to give the long time behavior of a precursor which is a stream of ions reflected by the electrostatic potential. An asymptotic expansion in terms of the small amplitude is used. The ratio of the ion temperature to the electron temperature is assumed small so that effects of the reflected ions appear as a correction to the Korteweg-de Vries equation which governs the wave propagation due to the nonresonant ions. The connection with the stationary solution is examined.