Conserved energies for the one dimensional Gross–Pitaevskii quation: low regularity case

We construct a family of conserved energies for the one dimensional Gross-Pitaevskii equation, but in the low regularity case (in [14] we have constructed conserved energies in the high regularity situation). This can be done thanks to regularization procedures and a study of the topological structure of the finite-energy space. The asymptotic (regularised conserved) phase change on the real line with values in $\mathbb{R}/2\pi\mathbb{Z}$ is studied. We also construct a conserved quantity, the renormalized momentum $H_1$ (see Theorem 1.3), on the universal covering space of the finite-energy space.