标题： 耦合色散 AB 系统在加权 Sobolev 初值下的长时间渐近行为 显示英文标题

标题： The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data

摘要： 在本工作中，我们采用$\bar{\partial}$-steepset 下降方法研究耦合色散 AB 系统的柯西问题，初始条件在加权 Sobolev 空间$H^{1,1}(\mathbb{R})$中， \begin{align*} \left\{\begin{aligned} &A_{xt}-\alpha A-\beta AB=0,\\ &B_{x}+\frac{\gamma}{2}(|A|^2)_t=0,\\ &A(x,0)=A_0(x),~~~~B(x,0)=B_0(x)\in H^{1,1}(\mathbb{R}). \end{aligned}\right. \end{align*}从耦合色散 AB 系统的 Lax 对出发，我们通过构建基本的黎曼-希尔伯特问题成功导出了耦合色散 AB 系统的解。 通过使用$\bar{\partial}$-steepset 下降方法，我们在没有离散谱的情况下表征了耦合色散 AB 系统解的长时间渐近行为。 我们的结果表明，与之前的结果相比，我们将长时间渐近解的精度从$O(t^{-1}\log t)$提高到$O(t^{-1})$。 显示英文摘要

摘要： In this work, we employ the $\bar{\partial}$-steepset descent method to study the Cauchy problem of the coupled dispersive AB system with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} \left\{\begin{aligned} &A_{xt}-\alpha A-\beta AB=0,\\ &B_{x}+\frac{\gamma}{2}(|A|^2)_t=0,\\ &A(x,0)=A_0(x),~~~~B(x,0)=B_0(x)\in H^{1,1}(\mathbb{R}). \end{aligned}\right. \end{align*} Begin with the Lax pair of the coupled dispersive AB system, we successfully derive the solutions of the coupled dispersive AB system by constructing the basic Riemann-Hilbert problem. By using the $\bar{\partial}$-steepset descent method, the long-time asymptotic behaviors of the solutions for the coupled dispersive AB system are characterized without discrete spectrum. Our results demonstrate that compared with the previous results, we increase the accuracy of the long-time asymptotic solution from $O(t^{-1}\log t)$ to $O(t^{-1})$.