# The initial-value problem for the cubic-quintic NLS with non-vanishing boundary conditions

## TL;DR

This paper studies the initial-value problem for a cubic-quintic nonlinear Schrödinger equation with non-vanishing boundary conditions, proving scattering for small initial data in a weighted Sobolev space.

## Contribution

It establishes a scattering result for the cubic-quintic NLS with non-zero boundary conditions, extending understanding of long-term behavior of solutions.

## Key findings

- Proves scattering for small initial data in weighted Sobolev spaces.
- Analyzes stability of constant solutions under the cubic-quintic NLS.
- Extends scattering theory to non-vanishing boundary conditions.

## Abstract

We consider the initial-value problem for 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 $|x|\to\infty$. Here $\alpha_1$, $\alpha_3$, $\alpha_5$ and $c$ are such that $\psi(x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $\psi(x)\to 1$ as $|x|\to\infty$, we study the associated initial-value problem for $u=\psi-1$ and prove a scattering result for small initial data in a weighted Sobolev space.