Vol. 9, No. 7, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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

(it + Δ)ψ = α1ψ α3|ψ|2ψ + α 5|ψ|4ψ

in three spatial dimensions in the class of solutions with |ψ(x)| c > 0 as |x|. 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 Hx1.

Keywords
final-state problem, wave operators, cubic-quintic NLS, nonvanishing boundary conditions
Mathematical Subject Classification 2010
Primary: 35Q55
Milestones
Received: 28 July 2015
Revised: 4 May 2016
Accepted: 9 July 2016
Published: 7 November 2016
Authors
Rowan Killip
Department of Mathematics
UCLA
Los Angeles, CA 90095-1555
United States
Jason Murphy
Department of Mathematics
University of California
Berkeley, CA 94720-3840
United States
Monica Visan
Department of Mathematics
UCLA
Los Angeles, CA 90095-1555
United States