Conserved energies for the one dimensional Gross-Pitaevskii equation

We prove the well-posedness results for 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 Hs regularities of the solutions. We establish a family of conserved energies for the one dimensional Gross-Pitaevskii equation, such that all 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.