The Final-State Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions


We construct solutions with prescribed scattering state to the cubic-quintic NLS (i∂t + Δ)ψ = α1ψ - α3|ψ|2ψ + α5|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.


Mathematics and Statistics


Killip was supported by NSF grant DMS-1265868. Murphy was supported by DMS-1400706. Visan was supported by NSF grant DMS-1161396. We are indebted to the Hausdorff Institute of Mathematics, which hosted us during our work on this project.

Keywords and Phrases

Cubic-quintic NLS; Final-state problem; Nonvanishing boundary conditions; Wave operators

International Standard Serial Number (ISSN)

2157-5045; 1948-206X

Document Type

Article - Journal

Document Version


File Type





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Publication Date

01 Nov 2016