Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case

TL;DR

This paper develops conserved energies for the one-dimensional Gross-Pitaevskii equation in low regularity settings, using regularization and topological analysis, extending previous high regularity results.

Contribution

It introduces a novel construction of conserved energies for low regularity solutions of the Gross-Pitaevskii equation, including a regularized phase change and a renormalized momentum.

Findings

Constructed conserved energies in low regularity setting

Analyzed the topological structure of the finite-energy space

Defined a conserved renormalized momentum on the universal cover

Abstract

We construct a family of conserved energies for the one dimensional Gross-Pitaevskii equation, but in the low regularity case (in \cite{KL} 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 $ \R/2\pi \Z$ is studied. We also construct a conserved quantity, the renormalized momentum $H_1$ (see Theorem \ref{thm:E1}), on the universal covering space of the finite-energy space.