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

Abstract

We construct solutions with prescribed scattering state to the cubic-quintic NLS

$$\left(i{\partial }_t+\Delta \right)\psi ={\alpha }_1\psi -{\alpha }_3|\psi {|}^2\psi +{\alpha }_5|\psi {|}^4\psi$$

in three spatial dimensions in the class of solutions with $\left|\psi \left(x\right)\right|\to c>0$ as $\left|x\right|\to \infty$. 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 weak topology on $H_x^1$.