论文标题
加权Sobolev空间中Camassa-Holm方程的Cauchy问题：长期和painlevé渐进性
The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics
论文作者
论文摘要
基于Deift-Zhou最陡峭的下降方法的$ \ overline \ part $ - 总体化，我们将Camassa-Holm（CH）方程式的长期和Painlevé渐近学扩展到解决方案中，并在加权Sobolev Space $ h^{4,2}（4,2}（\ Mathbbbbbbbbbbbb {\ mathbbbb {r r} r} r}）中，使用最初的数据使用初始数据。有了新的比例$（y，t）$和与初始值问题相关的RH问题，我们在不同的时空孤子区域中为CH方程的解决方案得出了不同的长期渐近扩展。 半平面$ \ {（y，t）： - \ infty <y <\ iftty，\ t> 0 \} $分为四个渐近区域：1。快速衰减区域，$ y/t \ in（ - \ infty，-1/4）$，带有错误$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {o}（o） 2。调制 - 苏里顿区域，$ y/t \ in（2，+\ infty）$，可以用带有残留错误$ \ MATHCAL {o}（t^{ - 1/2}）$的调制 - solitons来表征结果。 3. Zakhrov-Manakov地区，（0,2）$和$ Y/T \ in（-1/4,0）$。渐近近似的特征是分散术语和残留错误$ \ MATHCAL {O}（T^{ - 3/4}）$; 4。两个过渡区域，$ | y/t | \约2 $和$ | y/t | \大约-1/4 $，结果是通过painlevéii方程的解决方案来描述的，带有错误顺序$ \ mathcal {o}（t^{ - 1/2}）$。
Based on the $\overline\partial$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space $ H^{4,2}(\mathbb{R})$. With a new scale $(y,t)$ and a RH problem associated with the initial value problem,we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane $\{ (y,t): -\infty <y<\infty, \ t> 0\}$ is divided into four asymptotic regions: 1. Fast decay region, $ y/t \in(-\infty,-1/4)$ with an error $\mathcal{O}(t^{-1/2})$; 2. Modulation-solitons region, $y/t \in(2,+\infty)$, the result can be characterized with an modulation-solitons with residual error $\mathcal{O}(t^{-1/2 })$; 3. Zakhrov-Manakov region,$y/t \in(0,2)$ and $y/t \in(-1/4,0)$. The asymptotic approximations is characterized by the dispersion term with residual error $\mathcal{O}(t^{-3/4})$; 4. Two transition regions, $|y/t|\approx 2$ and $|y/t| \approx -1/4$, the results are describe by the solution of Painlevé II equation with error order $\mathcal{O}(t^{-1/2})$.
