The initial-value problem for the cubic-quintic NLS with non-vanishing boundary conditions

We consider the initial-value problem for the cubic-quintic NLS $$(i{\mathrm{\partial }}_t+\mathrm{\Delta })\psi ={\alpha }_1\psi -{\alpha }_3\mathrm{\mid }\psi {\mathrm{\mid }}^2\psi +{\alpha }_5\mathrm{\mid }\psi {\mathrm{\mid }}^4\psi$$ in three spatial dimensions in the class of solutions with $\mathrm{\mid }\psi (x)\mathrm{\mid }\to c>0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. Here ${\alpha }_1$, ${\alpha }_3$, ${\alpha }_5$ and $c$ are such that $\psi (x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $\psi (x)\to 1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, we study the associated initial-value problem for $u=\psi -1$ and prove a scattering result for small initial data in a weighted Sobolev space.