Stability Criterion for Dark Solitons.

We prove that the inequality $\ensuremath{\partial}P/\ensuremath{\partial}vl0$ is the stability criterion for dark solitons of the one-dimensional nonlinear Schr\"odinger equation with general nonlinearity. Here $v$ is the soliton's velocity, and $P$ renormalized momentum: $P\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(i/2)\ensuremath{\int}({\ensuremath{\psi}}_{x}^{*}\ensuremath{\psi}\ensuremath{-}{\ensuremath{\psi}}_{x}{\ensuremath{\psi}}^{*})(1\ensuremath{-}{\ensuremath{\rho}}_{0}/|\ensuremath{\psi}{|}^{2})\mathrm{dx}$.