Analysis of zero-frequency solutions of the pion dispersion equation in nuclear matter

In this paper we consider instability of nuclear matter which takes place when the frequencies of the collective excitations turn to zero. We investigate collective excitations with pion quantum numbers J^\pi=0^-. We study the dependence of zero-frequency solutions of the pion dispersion equation on the value of the spin-isospin quasiparticle interaction G'. The solutions of the pion dispersion equation describe the different types of the excitations in the matter, \omega_i(k). At the critical density \rho=\rho_c one of solutions of the definite type turns to zero: \omega_{i0}(k_c)=0. When \rho>\rho_c, the excitations \omega_{i0}(k) become amplified. It is shown that there is such a "transitional" value of G'=G'_{tr} that for G'<G'_{tr} the zero-frequency solutions belong to the type \omega_{sd} while for G'>G'_{tr} they pertain to the type \omega_c. The solutions of the type \omega_{sd} correspond to instability to small density fluctuations of the nuclear matter at G'\le -1. On the other hand, \omega_c is responsible for the "pion condensation" at G'\approx 2. For the stable nuclear matter the branches of solutions \omega_{sd}(k) and \omega_c(k) are located on the unphysical sheets of the complex plane of frequency.