The monotonicity conjecture and stability of solitons for the Cubic-Quintic NLS on R^3

TL;DR

This paper proves the stability and instability of solitons for the cubic-quintic nonlinear Schrödinger equation on R^3 at all frequencies, resolving the monotonicity conjecture and establishing a classification of solutions.

Contribution

It resolves the long-standing monotonicity conjecture and introduces a new variational framework for analyzing soliton stability in the cubic-quintic NLS.

Findings

Resolved the monotonicity conjecture for the cubic-quintic NLS.

Established a classification of normalized solutions.

Developed geometric analysis methods for spectral and variational problems.

Abstract

In this paper, we prove stability or instability of solitons for the cubic-quintic nonlinear Schrodinger equation at every frequency. The monotonicity conjecture raised by Killip, Oh, Pocovnicu and Visan is resolved. We introduce and solve a new cross-constrained variational problem. Then the uniqueness of the energy minimizers is proposed and shown. According to a spectral approach and variational arguments, we develop a set of geometric analysis methods. Correspondences between the soliton frequency and the prescribed mass are established. Classification of normalized solutions is given.