Kinetic description of collective oscillations in a system of parametric spin waves. III. Long-wave collective modes

Long-wave collective oscillations in a system of parametric spin waves of a ferromagnet are studied on the basis of the kinetic equations for the Wigner distribution functions of the magnons. A study is made of the linear response of the magnetization $\mathbf m$ to a change in the internal field $\mathbf h$ (in the case of deviation of a packet of parametric waves from a stationary state), and boundary-value problem for finding the eigenfrequencies of the collective modes in a sample of finite volume is formulated. It is shown that allowance for the influence of the boundary conditions leads to discreteness of the frequency spectrum of the collective modes in the region of wave numbers $q$ of the order of the reciprocal sample dimension $L^{-1}$. For a spherical sample, expressions are obtained for the discrete frequencies of long-wave collective modes corresponding to oscillations with longitudinal component $h_z$ of the internal field that does not depend on the coordinate $z$. Criteria for stability of stationary states of the ferromagnet with respect to the excitation of auto-oscillations are obtained.