Monotonicity Conjectures and Sharp Stability for Solitons of the Cubic-Quintic NLS on R^3

TL;DR

This paper resolves two key monotonicity conjectures for solitons in the cubic-quintic NLS on R^3, proves the uniqueness of energy minimizers, establishes sharp stability, and classifies normalized solutions.

Contribution

It provides complete proofs of two longstanding monotonicity conjectures, establishes the sharp stability of solitons, and offers a classification of normalized solutions for the cubic-quintic NLS.

Findings

Resolved frequency and mass monotonicity conjectures.

Proved uniqueness of energy minimizers.

Established sharp stability of solitons.

Abstract

This paper deals with the cubic-quintic nonlinear Schr\"{o}dinger equation on R^3. Two monotonicity conjectures for solitons posed by Killip, Oh, Pocovnicu and Visan are completely resolved: one concerning frequency monotonicity, and the other concerning mass monotonicity. Uniqueness of the energy minimizer is proved. Then sharp stability of the solitons is established. And classification of normalized solutions is first presented.