We consider the Gross-Pitaevskii equation on R-4 and the cubic-quintic nonlinear Schrodinger 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.
- NLS; Gross-Pitaevskii equation
- non-vanishing boundary condition
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