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

in three spatial dimensions in the class of solutions with $| ψ (x) | \to c > 0$ 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 weak topology on $H_{x}^{1}$ .

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

Vol. 9, No. 7, 2016

Rowan Killip, Jason Murphy and Monica Visan

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors