One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schr\"odinger Equation and Solution of Bogoliubov Equation in These Systems

In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schr"odinger equation, and solve the Bogoliubov equation of these systems. A map of spin-$n$ BEC is constructed from the $2^n$-dimensional spinor representation of irreducible tensor operators of $so(2n+1)$. Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.