On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schr\"odinger equations

TL;DR

This paper investigates the stability of solitary wave solutions in multi-dimensional cubic-quintic nonlinear Schrödinger equations, emphasizing the independence of different stability notions and providing numerical insights into their orbital stability.

Contribution

It clarifies the independence of standard stability notions and offers numerical analysis of ground state stability in radially symmetric cases.

Findings

Orbital stability notions are independent.

Numerical results support existing conjectures.

New conjectures are proposed based on simulations.

Abstract

We consider the nonlinear Schr\"odinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The core of this article is the notion of stability of solitary waves. We recall the two standard notions of orbital stability in the context of nonlinear Schr\"odinger equations, and show that they must be considered as independent from each other. We investigate numerically the notion of orbital stability of ground states in the radially symmetric case, confirming existing conjectures or leading to new ones.