Cauchy matrix structure and solutions of the spin-1 Gross-Pitaevskii equations

In this paper, we investigate the Cauchy matrix structure of the spin-1 Gross-Pitaevskii (GP) equations. By utilizing the Cauchy matrix approach, we derive a 2 x 2 matrix nonlinear Schrodinger (NLS) equation, which serves as an unreduced model for the spin-1 BEC system and allows solutions with explicit formulae. Then we provide suitable constraints which lead to reductions for obtaining the classical and nonlocal spin-1 GP equations and their solutions. Some obtained solutions, including one-soliton solution, two-soliton solution and double-pole solution, are analyzed and illustrated.