On Cauchy Problem for the Spin-1 Gross–Pitaevskii Equation: Soliton Resolution Conjecture and Asymptotic Analysis

We investigate the Cauchy problem for the spin-1 Gross–Pitaevskii (GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng et al. (Commun Math Phys 382:585–611, 2021) reported the long-time asymptotic result with error \(O(ln t/t)\) for the spin-1 GP equation for the case of the purely continuous spectrum. Based on the previous work, we conduct in-depth research on the soliton resolution conjecture and asymptotic analysis of the spin-1 GP equation. Compared with the previous work, we improve the asymptotic error accuracy from \(O(ln t/t)\) to \(O(t-3/4)\) . More importantly, through the \(∂ ̄\) -nonlinear steepest descent method and the Deift–Zhou’s nonlinear steepest descent method, we obtain effective asymptotic errors and successfully carry out a full asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including three cases: (i) coexistence of discrete and continuous spectrum; (ii) the purely continuous spectrum, as considered in the work of Geng et al. with error \(O(ln t/t)\) ; (iii) the purely discrete spectrum. For the case (i), the corresponding asymptotic approximations can be characterized by an N -solitons as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order \(O(t-3/4)\) . In the case (ii), we strictly prove that the solution of the spin-1 GP equation can be characterized by the soliton solution and an error term with \(O(t-3/4)\) . For the case (iii), we rigorously prove the localization of multiple degenerate soliton groups (DSGs), which is comprised of inseparable solitons with identical velocities, and calculate the long-time asymptotics for an arbitrary N -soliton solutions of the spin-1 GP equation. Finally, our results confirm the soliton resolution conjecture of the spin-1 GP equation and show that the soliton solutions of the spin-1 GP equation become a linear combination of multiple DSGs with different sizes.