solutions[@BaratoffBlackburnSchwartz; @Hakim], if the condensate velocity does not exceed a certain critical value. Near the critical point, which separates the stable branch and the unstable branch, it is known that many physical quantities obey saddle-node type scaling, such as an emission period of dark solitons[@Hakim; @PhamBrachet], an eigenvalue of a growing mode for the unstable solution[@PhamBrachet], and a transmission coefficient of linearized excitations[@Kovrizhin; @Kagan; @DanshitaYokoshiKurihara]. It is quite nontrivial that the time-reversible Hamiltonian system exhibits the scaling behaviors of saddle-node type, since this bifurcation is normally understood to emerge in time-irreversible phenomena. However, it is not easy to prove these properties analytically or exactly, because of the infinite dimensional character of the system.\
On the other hand, as another way to break the integrability, one can consider the generalization of the nonlinearity, that is what we will consider in the present paper. When the nonlinear term includes a competing interaction, the dark soliton is no longer always stable. One typical example of an unstable dark soliton is a “bubble” in a cubic-quintic NLS (CQNLS) system[@BarashenkovMakhankov; @BarashenkovPanova]. (See also [@JYang].) The most general criterion for the stability of the dark soliton has been shown by Barashenkov[@Barashenkov], and the critical velocity of 