Existence of small semi-vortex solutions for the cubic nonlinear Schr\"{o}dinger system with Rashba type Spin-Orbit coupling on $\mathbb{R}^2$

TL;DR

This paper proves the existence of small semi-vortex solutions for a specific nonlinear Schrödinger system with Spin-Orbit coupling, using energy minimization under a small mass constraint.

Contribution

It provides rigorous mathematical proofs for the existence of semi-vortex solutions, previously known only in physics literature, via variational methods.

Findings

Existence of small semi-vortex solutions established.

Existence of small ground states proved.

Mathematical validation of physics predictions achieved.

Abstract

We consider the cubic nonlinear Schr\"{o}dinger system with Rashba type Spin-Orbit coupling (SOC) on $\mathbb{R}^2$, which is also called the Gross--Pitaevskii equation with SOC. The system describes SO-coupled spinor BEC in physics. In the literature of physics, the existence of small semi-vortex solutions and small ground state is known. In the present paper, we give their mathematical proofs by finding minimizers of the energy under small mass constraint.