We construct solutions with prescribed scattering state to the cubic-quintic
NLS
in three spatial dimensions in the class of solutions with
as
. This models
disturbances in an infinite expanse of (quantum) fluid in its quiescent state — the limiting
modulus
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
weak topology on
.