Complete classification of solutions for spin-1 Bose-Einstein condensates in R

Spin-1 Bose-Einstein condensates provide a fundamental platform for exploring quantum magnetism and coherent spin dynamics. Their three Zeeman components are coupled by spin-exchange interactions under conserved particle number and magnetization, giving rise to rich magnetic phases and nonlinear spin structures; a rigorous classification of stationary states therefore clarifies the possible spin configurations and their physical origins. In this paper, as a continuation of our previous work, we study the classification of solution for the spin-1 Bose-Einstein condensates in R, by applying the non-standard Hirota's bilinear method, the determinant formula and the general left-uniqueness strategy, we present some novel explicit solution and give the complete classification of the solutions. In comparison with the classical Hirota's method, the non-standard Hirota's method needs to introduce auxiliary functions to construct Hirota's bilinear form, which essentially adds additional constraint on the wave functions, so that it possible to deal with more complex equations. As far as we know, this is the first systematic study the complete classification of solutions to the spin-1 Bose-Einstein condensates. Our classification results yields experimentally testable predictions for spin populations, relative phases, and magnetic textures