On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential

TL;DR

This paper investigates the existence and stability of ground states for the 2D nonlinear Schrödinger equation with combined nonlinearities and a harmonic potential, providing new insights into the parameter regimes for stable standing waves.

Contribution

It adapts the fundamental frequency solutions method to the 2D case with potential, characterizing and proving the orbital stability of ground states in this setting.

Findings

Characterization of the set of fundamental frequency standing waves.

Proof of orbital stability for these standing waves.

Extension of the method to the 2D case with harmonic potential.

Abstract

We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the twodimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.