标题： 耦合色散AB系统的长时间渐近行为在低正则性空间中 显示英文标题

标题： Long-time Asymptotic Behavior of the coupled dispersive AB system in Low Regularity Spaces

摘要： 在本文中，我们主要研究耦合色散AB系统在加权Sobolev初始数据下的解的长时间渐近行为，该系统可以通过Dbar最陡下降方法产生孤子解。基于Lax对的谱分析，耦合色散AB系统的Cauchy问题被转化为一个黎曼-希尔伯特问题，并通过消失引理证明了其解的存在性和唯一性。 平稳相位点在长时间渐近行为中起着重要作用。 我们证明，在任意固定时间锥$\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} \mid x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\}$中，耦合色散AB系统解的长时间渐近行为可以表示为离散谱上的$N(\mathcal{I})$个孤子，连续谱上的主项$\mathcal{O}(t^{-1 / 2})$以及允许的剩余项$\mathcal{O}(t^{-3 / 4})$。 显示英文摘要

摘要： In this paper, we mainly investigate the long-time asymptotic behavior of the solution for the coupled dispersive AB system with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method.Based on the spectral analysis of Lax pair, the Cauchy problem of the coupled dispersive AB system is transformed into a Riemann-Hilbert problem, and its existence and uniqueness of the solution is proved by the vanishing lemma. The stationary phase points play an important role in the long-time asymptotic behavior. We demonstrate that in any fixed time cone $\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} \mid x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\}$, the long-time asymptotic behavior of the solution for the coupled dispersive AB system can be expressed by $N(\mathcal{I})$ solitons on the discrete spectrum, the leading order term $\mathcal{O}(t^{-1 / 2})$ on the continuous spectrum and the allowable residual $\mathcal{O}(t^{-3 / 4})$.