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

TL;DR

This paper maps one-dimensional integrable spinor Bose-Einstein condensates to a matrix nonlinear Schrödinger equation and solves the associated Bogoliubov equations using squared Jost functions, advancing understanding of their integrable structure.

Contribution

It introduces a novel mapping from spinor BECs to matrix nonlinear Schrödinger equations and provides explicit solutions to the Bogoliubov equations for these systems.

Findings

Mapping from spin-$n$ BEC to matrix NLS established

Solutions to Bogoliubov equations obtained using squared Jost functions

Framework enables analysis of excitations in integrable spinor BECs

Abstract

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.