The 1D Gross-Pitaevskii equation with a traveling delta potential is globally well-posed in the energy space, and the higher-density stationary state below the critical velocity is orbitally stable.
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On the Gross-Pitaevskii model with a moving impurity: Cauchy problem and superfluidity criterion
The 1D Gross-Pitaevskii equation with a traveling delta potential is globally well-posed in the energy space, and the higher-density stationary state below the critical velocity is orbitally stable.