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

Abstract

We construct solutions with prescribed scattering state to the cubic-quintic NLS $$(i{\partial }_t+\hat{I}”)\ddot{I}ˆ=\hat{I}{\pm }_1\ddot{I}ˆ-\hat{I}{\pm }_3|\ddot{I}ˆ{|}^2\ddot{I}ˆ+\hat{I}{\pm }_5|\ddot{I}ˆ{|}^4\ddot{I}ˆ$$ in three spatial dimensions in the class of solutions with $|\ddot{I}ˆ(x)|\to c>0$ 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 $H_x^1$.

46 pages