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T. Gallay and D.E. Pelinovsky

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

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Journal of Differential Equations 258 (2015), 3639-3660

**Abstract:**

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}(R).

**Keywords**:

Nonlinear Schrodinger equation; the black soliton; orbital stability;
conserved quantities.