The final-state problem for the cubic-quintic NLS with non-vanishing boundary conditions

We construct solutions with prescribed scattering state to 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 $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. 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 $H_x^1$.