Solitons and scattering for the cubic-quintic nonlinear Schr\"odinger equation on $\mathbb{R}^3$

TL;DR

This paper studies the cubic-quintic nonlinear Schrödinger equation on three-dimensional space, analyzing solitons, their stability, and scattering behavior, revealing new insights into soliton rescaling and limitations of virial methods.

Contribution

It characterizes ground-state solitons, their stability, and introduces the concept of rescaled solitons, demonstrating their role in scattering and limitations of virial-based approaches.

Findings

Characterized the shape of the mass/energy curve for solitons.

Identified the kernel of the linearized operator around solitons.

Proved scattering for solutions in a specific mass/energy region, bounded by solitons and rescaled solitons.

Abstract

We consider the cubic-quintic nonlinear Schr\"odinger equation: \[ i\partial_t u = -\Delta u - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region $\mathcal{R}$ of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.