耦合非线性薛定谔方程多孤子解的渐近稳定性：基于$\bar{\partial}$-方法
Asymptotic Stability of Multi-Solitons for Coupled Nonlinear Schrödinger Equations via the $\bar{\partial}$-Method
- South China University of Technology（华南理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文基于$\bar{\partial}$-最速下降法，建立了聚焦耦合非线性薛定谔方程在加权索伯列夫空间中的长时间渐近行为，将误差估计改进至$t^{-3/4}$阶，并证明了多孤子解的渐近稳定性及碰撞规律。
AI中文摘要：
聚焦耦合非线性薛定谔（CNLS）方程的黎曼-希尔伯特问题是在相应的$3\times3$矩阵谱问题基础上建立的。我们借助达布变换去除了初始RHP的离散谱。基于$\bar{\partial}$-最速下降法，我们建立了在加权索伯列夫空间中初始条件下CNLS方程解的长时间渐近行为。与改进的非线性最速下降法相比，我们将误差估计改进至$\mathcal{O}\left(t^{-3/4}\right)$阶。此外，我们获得了CNLS方程多孤子解的渐近稳定性，并从杨-巴克斯特映射的角度分析了多孤子碰撞的规律。
英文摘要：
The Riemann-Hilbert problem for the focusing coupled nonlinear Schrödinger (CNLS) equation is formulated on the basis of the corresponding $3\times3$ matrix spectral problem. We remove the discrete spectrum of initial RHP with the aid of Darboux transformations. Based on the $\bar{\partial}$-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 $\mathcal{O}\left(t^{-3/4}\right)$. 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.
