The Alfvén Shock Waves in a Cold Plasma in the Presence of Effective Electron-Ion Collisions

Nonlinear Alfvén waves in a cold plasma in the presence of effective electron-ion collisions are investigated by means of a nonlinear perturbation method. To the lowest order of perturbation, it is found that the system of equations for the Alfvén waves can be reduced to the following simple dispersive-dissipative equation for f (which is proportional to perturbed part of normalized density): \(∂ f/∂τ-3/2μAf²∂ f/∂ξ-μA{∂³f}{∂ξ³}-ν∂/∂ξ\{∂ f/∂ξ+f∫-∞ξf²dξ\}{=}0\), where ξ and τ are, respectively, stretched space and time coordinates, while µ A and ν denote, respectively, dispersion and dissipation parameters. It is shown that this equation has solution representing either oscillatory (dispersion-dominant case) or monotonous (dissipation-dominant case) Alfvén shock waves.