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

TL;DR

This work resolves two monotonicity conjectures for solitons in the cubic-quintic NLS on $\mathbb{R}^3$ by establishing strict monotonicity of the frequency-related function $\beta(\omega)$ and the mass $M(P_\omega)$ across the admissible frequency range, and it proves the uniqueness of the energy minimizer for the associated variational problems. It builds a comprehensive variational framework linking ground states $P_\omega$, rescaled solitons $R_\omega$, and their normalized-solution counterparts, and uses this to derive sharp orbital stability thresholds via Grillakis–Shatah–Strauss analysis. The paper also provides a complete classification of normalized solutions, identifying when multiple solitons coexist for a given mass and when uniqueness holds. These results illuminate the global dynamics of the defocusing-cubic perturbation of the energy-critical NLS, including stability/instability regimes and the precise mass–frequency correspondences that underpin the soliton structure.

Abstract

This paper deals with the cubic-quintic nonlinear Schrödinger 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.

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 2.4
• Proposition 3.1