论文标题
具有加权Sobolev初始数据的耦合色散AB系统的长期渐近行为
The long-time asymptotic behaviors of the solutions for the coupled dispersive AB system with weighted Sobolev initial data
论文作者
论文摘要
在这项工作中，我们采用$ \ bar {\ partial} $ - 陡峭的下降方法来研究具有加权Sobolev Space的初始条件$ H^{1,1}（\ MATHBB {r}）$， \ begin {Align*} \ left \ {\ begin {aLigned}＆a_ {xt}-αA-βab= 0，\\＆b_ {x}+\fracγ{2}（| a | a | a |^2）_t = 0， h^{1,1}（\ mathbb {r}）。 \ end {Aligned} \ right。 \ end {Align*}从耦合的分散AB系统的宽松对开始，我们通过构造基本的Riemann-Hilbert问题成功地得出了耦合的色散AB系统的解决方案。通过使用$ \ bar {\ partial} $ - 陡峭的下降方法，可以将耦合分散AB系统解决方案的长期渐近行为表征，而无需离散光谱。我们的结果表明，与先前的结果相比，我们将长期渐近解决方案的精度从$ O（t^{ - 1} \ log t）$提高到$ O（t^{ - 1}）$。
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}-αA-βAB=0,\\ &B_{x}+\fracγ{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 $O(t^{-1})$.
