Sharp Stability of Solitons for the Cubic-Quintic NLS on R^2

TL;DR

This work analyzes the two-dimensional cubic-quintic NLS $i\partial_t \varphi + \Delta \varphi + |\varphi|^2\varphi - |\varphi|^4\varphi = 0$ on $\mathbb{R}^2$, addressing orbital stability of solitons across all frequencies. By developing a new variational framework with a family of functionals $F_\alpha$ and associated minimizers $Q_\alpha$, it links these minimizers to ground states $P_\omega$ and proves strict monotonicity of key quantities, enabling a one-to-one mass-frequency correspondence. The authors establish sharp orbital stability for solitons at every frequency with mass above the ground-state threshold, and, for the first time, provide a complete classification of normalized solutions via the mass–frequency map $m \mapsto \omega$. The results resolve open questions in the literature and introduce a universal variational approach applicable to non-scale-invariant soliton problems.

Abstract

This paper concerns with the cubic-quintic nonlinear Schrödinger equation on R^2. A family of new variational problems related to the solitons are introduced and solved. Some key monotonicity and uniqueness results are obtained. Then the orbital stability of solitons at every frequency are proved in terms of the Cazenave and Lions' argument. And classification of normalized ground states is first presented. Our results settle the questions raised by Lewin and Rota Nodari as well as Carles and Sparber.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1