Orbital stability in the cubic defocusing NLS equation: II. The black   soliton
  Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$. 
Thierry Gallay and Dmitry Pelinovsky
math.AP