Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr\"odinger equations with non-vanishing boundary conditions

TL;DR

This paper proves the global well-posedness and unconditional uniqueness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr"odinger equations with non-vanishing boundary conditions, viewed as perturbations of the energy-critical NLS.

Contribution

It establishes the global well-posedness and unconditional uniqueness of these equations in their energy spaces, extending understanding of their mathematical properties.

Findings

Global well-posedness in energy spaces

Unconditional uniqueness in energy spaces

Perturbation approach to energy-critical NLS

Abstract

We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.