Three-dimensional stability of solitary kinetic Alfvé waves and ion-acoustic waves

Starting from a set of equations that lead to a linear dispersion relation coupling kinetic Alfvén waves and ion-acoustic waves, three-dimensional KdV equations are derived for these waves. These equations are then used to investigate the three-dimensional stability of solitary kinetic Alfvén waves and ion-acoustic waves by the small-k perturbation expansion method of Rowlands and Infeld. For kinetic Alfvén waves it is found that there is instability if the direction of the plane-wave perturbation lies inside a cone, and the growth rate of the instability attains a maximum when the direction of the perturbation lies in the plane containing the external magnetic field and the direction of propagation of the solitary wave.