#### Vol. 9, No. 7, 2016

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions

### Rowan Killip, Jason Murphy and Monica Visan

Vol. 9 (2016), No. 7, 1523–1574
DOI: 10.2140/apde.2016.9.1523
##### 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 $|\psi \left(x\right)|\to c>0$ 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 weak topology on ${H}_{x}^{1}$.

##### Keywords
final-state problem, wave operators, cubic-quintic NLS, nonvanishing boundary conditions
Primary: 35Q55