AI 中文总结
研究色散波区域中良好Boussinesq方程解的渐近稳定性，运用Dbar最速下降法于Riemann-Hilbert问题，得到特定阶的渐近展开，扩展初始数据空间，证明了解在该区域的渐近稳定性。
AI 中文摘要
本文研究了色散波区域中良好Boussinesq方程解的渐近稳定性，此时与初始数据相关的反射系数属于加权Sobolev空间。将Dbar最速下降法应用于Riemann-Hilbert问题，得到了直至最优误差为\(\mathcal{O}(t^{-3/4})\)阶的解的长时间渐近展开。与之前结果相比，将初始数据从快速衰减的Schwartz空间扩展到加权Sobolev空间，并证明了解在色散波区域的渐近稳定性。
英文摘要
This work studies the asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region when the reflection coefficients associated with the initial data belong to weighted Sobolev space. The Dbar-steepest descent method is applied to the Riemann-Hilbert problem and the long-time asymptotic expansion of the solution are obtained up to an optimal error of order $\mathcal{O}(t^{-3/4})$. Compared with previous results, we extend the initial data from the rapidly decaying Schwartz space to a weighted Sobolev space, and prove the asymptotic stability of the solution in dispersive wave region.
