Black solitons are orbitally stable in the energy space when the derivative of momentum with respect to speed is negative at zero speed.
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Proves the VK criterion applies to quasilinear GP dark solitons, implying stability for weak quasilinear terms and conditional stability (fast stable, slow unstable) for stronger terms due to an energy-momentum cusp.
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Orbital stability of black solitons for quasilinear Schr\"odinger equations with nonzero conditions at infinity
Black solitons are orbitally stable in the energy space when the derivative of momentum with respect to speed is negative at zero speed.
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Stability and instability of the quasilinear Gross--Pitaevskii dark solitons
Proves the VK criterion applies to quasilinear GP dark solitons, implying stability for weak quasilinear terms and conditional stability (fast stable, slow unstable) for stronger terms due to an energy-momentum cusp.