  We construct solutions with prescribed scattering state to the cubic-quintic
NLS $$ (i\partial_t+\Delta)\psi=\alpha_1 \psi-\alpha_{3}\vert \psi\vert^2
\psi+\alpha_5\vert \psi\vert^4 \psi $$ in three spatial dimensions in the class
of solutions with $|\psi(x)|\to c >0$ as $|x|\to\infty$. This models
disturbances in an infinite expanse of (quantum) fluid in its quiescent state
--- the limiting modulus $c$ corresponds to a local minimum in the energy
density.
  Our arguments build on work of Gustafson, Nakanishi, and Tsai on the
(defocusing) Gross--Pitaevskii equation. The presence of an energy-critical
nonlinearity and changes in the geometry of the energy functional add several
new complexities. One new ingredient in our argument is a demonstration that
solutions of such (perturbed) energy-critical equations exhibit continuous
dependence on the initial data with respect to the \emph{weak} topology on
$H^1_x$.
