B. Nonlinear Modulation of Plasma Waves

The self-trapping and instabilities of modulated whistler waves propagating along a uniform magnetic field were first studied by Taniuti arid Washimi.l) Using the fluid model for the cold plasma, they derived a dispersive nonlinear equation (the nonlinear Schrodinger equation) to describe a slow modulation of a carrier wave of small but finite amplitude. This equation takes the same form as the equation appearing in the study of self-focusing in nonlinear optics. They then showed that the solitary wave, that is, envelope soliton exists while the plane wave is unstable against a modulation of amplitude and phase. Mizutani and Taniuti2) extended the theory to oblique propagation of stationary waves to obtain the envelope soliton, and later Mizutani3) derived the nonlinear Schrodinger equation for nonstationary oblique propagation and showed the modula~ional instability. These problems were also investigated by Pataraya.4) However, these works are concerned with the propagation of the ·carrier wave at those frequencies for which the group velocity is equal to the phase velocity. The extension to propagation at an arbitrary frequency is straightforward, and this was carried out, using the method of Taniuti and Yajima,5) by Hasegawa6) for the case of propagation along the magnetic field for all the frequencies of whistler waves and also ion-cyclotron waves. Further, Kako7) has investigated the modulation of carrier waves of arbitrary frequencies for all angles of propagation, so that the results obtained earlier are special cases. In this paper, we present an investigation of nonlinear wave