Self-modulation of ion Bernstein waves

A self-modulation equation for coherent ion Bernstein waves is derived by a multiple-time-scale, perturbative expansion of the Vlasov equation. The dominant nonlinearity acting on the waves is shown to be the nonlinear ion gyrofrequency shift. The waves obey a version of the nonlinear Schrödinger equation with the dispersive terms in the three spatial directions having, in general, differing signs. If the modulation is independent of one spatial variable, the elliptic or hyperbolic two-dimensional nonlinear Schrödinger equation results. In contrast to the elliptic case which exhibits collapse, wave energy always disperses if initially localized in the hyperbolic case.