On the Gross-Pitaevskii model with a moving impurity: Cauchy problem and superfluidity criterion

We study the one-dimensional Gross-Pitaevskii equation with a traveling delta potential and non-zero conditions at infinity.

This model describes the effect of a moving impurity in a quantum fluid.

Firstly, we show that the associated Cauchy problem is globally well-posed in the energy space.

This requires the definition of a conserved energy, which involves the notion of renormalized momentum.

Secondly, we study the existence and stability of stationary states in a co-moving reference frame.

It is known that there exists an impurity-dependent critical velocity above which no stationary state exists.

For velocities below the critical one, two different stationary states appear.

We show the orbital stability of the one with higher minimal density.