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

Abstract

We consider the initial-value problem for the cubic-quintic NLS \[ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ\] in three spatial dimensions in the class of solutions with $|\ddot{I}ˆ(x)|\to c>0$ as $|x|\to \infty$. Here $\hat{I}{\pm }_1$, $\hat{I}{\pm }_3$, $\hat{I}{\pm }_5$ and $c$ are such that $\ddot{I}ˆ(x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $\ddot{I}ˆ(x)\to 1$ as $|x|\to \infty$, we study the associated initial-value problem for $u=\ddot{I}ˆ-1$ and prove a scattering result for small initial data in a weighted Sobolev space.

57 pages