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



Rowan Killip (Department of Mathematics, University of California at Los Angeles)

Tadahiro Oh (Department of Mathematics, Princeton University)

Oana Pocovnicu (Department of Mathematics, Princeton University)

Monica Visan (Department of Mathematics, University of California at Los Angeles)


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.


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

2010 Mathematics Subject Classification


Published 15 March 2013