具有加权Sobolev初值的Harry Dym方程的孤子解
Soliton resolution for the Harry Dym equation with weighted Sobolev initial data
摘要：
在加权Sobolev空间$H^{1,1}(\mathbb{R})$中，对初始条件建立了Harry Dym方程的孤子解.结合非线性最速下降法和$\bar{\partial}$ -导数条件，得到了当$\frac{y}{t}<-ε(ε>0)$时，解$q(x,t)$在任意固定锥\begin{equation} C\left(y_{1}, y_{2}, v_{1}, v_{2}\right)=\left\{(y, t) \in R^{2} \mid y=y_{0}+v t, y_{0} \in\left[y_{1}, y_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\} \end{equation}中的长时间渐近展开式直到一个$\mathcal{O}(t^{-1})$阶的残差.结果表明，长时间的渐近行为可以用离散谱上的$N(I)$孤子来描述，其参数由通过圆锥时的局域孤子-孤子相互作用和连续谱上的孤子-辐射相互作用的第二项调制.
英文摘要：
The soliton resolution for the Harry Dym equation is established for initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})$. Combining the nonlinear steepest descent method and $\bar{\partial}$-derivatives condition, we obtain that when $\frac{y}{t}<-ε(ε>0)$ the long time asymptotic expansion of the solution $q(x,t)$ in any fixed cone \begin{equation} C\left(y_{1}, y_{2}, v_{1}, v_{2}\right)=\left\{(y, t) \in R^{2} \mid y=y_{0}+v t, y_{0} \in\left[y_{1}, y_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\} \end{equation} up to an residual error of order $\mathcal{O}(t^{-1})$. The expansion shows the long time asymptotic behavior can be described as an $N(I)$-soliton on discrete spectrum whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone and the second term coming from soliton-radiation interactionson on continuous spectrum.
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