Threshold solutions for the 3d cubic-quintic NLS

TL;DR

This paper analyzes the boundary behavior of solutions to the 3D cubic-quintic nonlinear Schrödinger equation, classifying solutions near soliton thresholds and identifying special scattering and convergence phenomena.

Contribution

It provides a classification of boundary solutions for the cubic-quintic NLS, including the identification of a unique special solution with mixed scattering and convergence properties.

Findings

Solutions on the soliton boundary either scatter or converge exponentially to a soliton.

A special solution exists that scatters in one time direction and converges in the other.

The boundary behavior is characterized by ground state solitons and their rescalings.

Abstract

We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and by certain rescalings thereof. We classify the possible behaviors of solutions on the part of the boundary attained solely by solitons. In particular, we show that non-soliton solutions either scatter in both time directions or coincide (modulo symmetries) with a special solution, which scatters in one time direction and converges exponentially to the soliton in the other.