Taking self-gravity into account, we investigated the stability of the interstellar medium in which the circularly polarized Alfvén wave travels with large amplitude. The interstellar medium suffers from two types of instabilities: the Jeans instability and the parametric instability. Using a linear stability analysis, we show both the effect of the Alfvén wave on the Jeans instability and the effect of self-gravity on the parametric instability. The Alfvén wave suppresses the Jeans instability, having effective pressure which acts against gravity. The effective pressure is proportional to the wave energy density and depends on the wavelength of the Alfvén wave and Alfvén speed. The parametric instability transforms the Alfvén wave into other magnetohydrodynamical waves. We followed the nonlinear growth of the instabilities with a newly developed computation code. The development of the computation code is summarized in Appendix A. Based on a linear stability analysis and numerical simulation, we discuss the magnetohydrodynamical evolution of the interstellar medium.
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