Asymptotic Stability of Multi-Solitons for Coupled Nonlinear Schrödinger Equations via the $\bar{\mathrm{\partial }}$-Method

The Riemann-Hilbert problem for the focusing coupled nonlinear Schrödinger (CNLS) equation is formulated on the basis of the corresponding $3\times 3$ matrix spectral problem. We remove the discrete spectrum of initial RHP with the aid of Darboux transformations. Based on the $\bar{\mathrm{\partial }}$-steepest descent method, we establish the long-time asymptotic behavior of solutions to the CNLS equation for initial condition in the weighted Sobolev space. Compared to the improved nonlinear steepest descent method, we improve the error estimate up to order $\mathcal{O}\left(t^{-3\mathrm{/}4}\right)$. Furthermore, we obtain the asymptotic stability of multi-soliton solutions for CNLS equation and analyze the law of multi-soliton collision in the view-point of Yang-Baxter map.