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

TL;DR

This paper establishes the orbital stability of solitons for the cubic-quintic nonlinear Schrödinger equation on R^2, introducing new variational problems, proving key properties, and classifying ground states to answer open questions.

Contribution

It introduces new variational problems related to solitons, proves their stability, and classifies ground states for the cubic-quintic NLS on R^2, addressing prior open questions.

Findings

Orbital stability of solitons at all frequencies is proved.

New variational problems related to solitons are solved.

Classification of normalized ground states is provided.

Abstract

This paper concerns with the cubic-quintic nonlinear Schr\"{o}dinger 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.