As has been shown previously for the case of a cold plasma, the structure and propagation of hydromagnetic waves of finite but small amplitude can be described by the Korteweg-de Vries equation. This equation can be derived from the set of the hydromagnetic equations for a collisionless cold plasma by applying certain scaling relations between the pertinent field variables. For a cold plasma, the hydromagnetic transport equations are equivalent to the Vlasov equations because the distribution functions of the particle velocities are delta functions. In the case of a warm plasma, the initial distribution functions are Maxwellian. Starting from the Vlasov equations for the electron and ion gas of the plasma, the procedure as applied in the case of a zero temperature plasma leads, under known conditions, to a generalized time-dependent Korteweg-de Vries equation, with coefficients depending on the temperature of either plasma component and on the angle between the direction of wave propagation and the initial magnetic field. Charge separation effects and displacement currents appear to be negligible to lowest order.
