Conserved energies for the one dimensional Gross-Pitaevskii equation

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$^{8}$ 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.