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Combining the usual energy functional with a higher-order conserved quantityoriginating from integrability theory, we show that the black soliton is alocal minimizer of a quantity that is conserved along the flow of the cubicdefocusing NLS equation in one space dimension. This unconstrained variationalcharacterization gives an elementary proof of the orbital stability of theblack soliton with respect to perturbations in $H^2(\mathbb{R})$.

Orbital stability in the cubic defocusing NLS equation: II. The black soliton