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# Mathematical Research Letters

## Volume 19 (2012)

### Number 5

### Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions

Pages: 969 – 986

DOI: https://dx.doi.org/10.4310/MRL.2012.v19.n5.a1

#### Authors

#### Abstract

We consider the Gross–Pitaevskii equation on $\mathbb{R}^4$ and the cubic-quintic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ 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.

#### Keywords

NLS; Gross–Pitaevskii equation, non-vanishing boundary condition

#### 2010 Mathematics Subject Classification

35Q55

Published 15 March 2013