无穷维Vey定理及其对KdV的应用
Vey theorem in infinite dimensions and its application to KdV
摘要：
考虑Hilbert空间$H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$中的可积无穷维Hamilton系统，其积分为$I_1, I_2,...$，可记为$I_j={1/2}|F_j|^2$，其中$F_j:H\to \R^2$，$F_j(0)=0$为$j=1,2,...$ .我们假设映射$F_j$定义了解析同胚$F=(F_1,F_2,...):H\to H$的芽，使得dF（0）=id $, $（F-id）$ is a $ κ $-smoothing map ($ κ \geq 0 $) and some other mild restrictions on $ F $ hold. Under these assumptions we show that the maps $ F_j $ may be modified to maps $ F_j^ \prime $ such that $ F_j-F_j^ \prime =O（|u| ^2）$ and each $ \frac 12| F'_j| ^2 $ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $ F^ \prime：H \to H $, the germ $（F^ \prime -id）$ is $ κ $-smoothing, and each $ I_j $ is an analytic function of the vector $（\frac 12| F'_j| 1）$. Next we show that the theorem with $ κ=1$适用于KdV方程。这意味着在函数空间KdV的原点附近，Birkhoff标准形和积分变换的形式是“单位元加1-光滑解析映射”。
英文摘要：
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $κ$-smoothing map ($κ\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $κ$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $κ=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.
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