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

We consider the cubic nonlinear Schrödinger system with Rashba type Spin-Orbit Coupling (SOC) on $\mathbb{R}^2$.

The system describes SO-coupled spinor BEC in physics.

In the literature of physics, the small semi-vortex solutions, small ground state, and the so-called mixed mode, which are mixture of semi-vortex solutions, are investigated.

The semi-vortex solutions cause from the resonance on the essential spectrum of the linear operator.

In the present paper, we give mathematical proofs of the existence of the semi-vortex and the ground state by finding minimizers of the energy under small mass constraint based on concentration compactness argument.

Moreover, we also discuss the mixed modes in the case where all the coefficients of the nonlinear terms are equal.