Cubic Nonlinear Schrödinger Systems and Orthonormal Fermionic Ground States in $\mathbb R$

We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=μ_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots,u_N)\in H^1(\mathbb{R})^N$, $μ_1\leqμ_2\leq\cdots\leqμ_N<0$, $1<p<3$ and $N\geq 2$ are arbitrary. For th