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

Herbert Koch; Xian Liao

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arXiv · 2204.06293

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

Herbert Koch·Xian Liao

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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.

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Herbert Koch, Xian Liao. 2022-04-13. Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case. https://arxiv.org/abs/2204.06293

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