Conserved energies for the one dimensional Gross-Pitaevskii equation

TL;DR

This paper proves global well-posedness for the 1D Gross-Pitaevskii equation in an energy space and introduces a family of conserved energies that remain invariant over time, also conserved by related flows.

Contribution

It introduces a new metric and conserved energies for the 1D Gross-Pitaevskii equation, extending understanding of its long-term behavior and invariants.

Findings

Global well-posedness in energy space established

A new family of conserved energies introduced

Conservation extends to complex modified KdV flow

Abstract

We prove the global-in-time well-posedness of the one dimensional Gross-Pitaevskii equation in the energy space, which is a complete metric space equipped with a newly introduced metric and with the energy norm describing the $H^s$ regularities of the solutions. We establish a family of conserved energies for the one dimensional Gross-Pitaevskii equation, such that the energy norms of the solutions are conserved globally in time. This family of energies is also conserved by the complex modified Korteweg-de Vries flow.