Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma

The hydromagnetic waves with small but finite amplitude in a cold collision-free plasma are investigated by using a nonlinear perturbation method. In the lowest order of perturbation, we can show that the system of equations for the magneto-acoustic wave propagating along a `critical' direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one. An extension of the problem to more general dispersive system is also made. On the other hand, the system of equations for the Alfvén wave is reduced to a modified Korteweg-de Vries equation in the sense that the non-linear term f ∂ f /∂ξ in the Korteweg-de Vries equation is replaced by f 2 ∂/∂ξ. In the case of steady propagation this equation can be integrated to give a solution in closed form, which exhibits a solitary wave. Two kinds of solitary wave (both compressive and rarefied) are found to be possible.