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

TL;DR

This paper constructs solutions with prescribed scattering states for the cubic-quintic nonlinear Schrödinger equation in three dimensions, modeling disturbances in a quantum fluid with non-vanishing boundary conditions, and addresses new complexities due to energy-critical nonlinearities.

Contribution

It extends existing methods to handle energy-critical nonlinearities and non-vanishing boundary conditions in the cubic-quintic NLS, demonstrating continuous dependence on initial data in the weak topology.

Findings

Successfully constructed solutions with prescribed scattering states.

Established continuous dependence on initial data in the weak topology.

Addressed complexities arising from energy-critical nonlinearities.

Abstract

We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+\Delta)\psi=\alpha_1 \psi-\alpha_{3}\vert \psi\vert^2 \psi+\alpha_5\vert \psi\vert^4 \psi $$ in three spatial dimensions in the class of solutions with $|\psi(x)|\to c >0$ 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 \emph{weak} topology on $H^1_x$.