Abstract
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 $I_j=\frac{1}{2}|F_j|^2$, where $F_j:H\rightarrow \R^2$, $F_j(0)=0$ for $j=1,2,....$ We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\rightarrow H$, such that $dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) 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 such that $F_j-$F ’j$=O(|u|^2)$ and each 1/2|F ’j|$^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism F’$: H\rightarrow H$, the germ (F’-id) is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector (1/2|F’j|$^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ 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'.
Highlights
R2, Fj(0) = 0 for j = 1, 2, . . . . We assume that the maps Fj define a germ of an analytic diffeomorphism F = (F1, F2, . . . ) : H → H, such that dF (0) = id, (F − id) is a κ-smoothing map (κ ≥ 0) and some other mild restrictions on F hold. Under these assumptions we show that the maps Fj may be modified to maps Fj such that Fj − Fj = O(|u|2)
Vey proved a local version of the Liouville-Arnold theorem which we state for the case of an elliptic singular point. 1
[Eli84, Eli90] Eliasson suggested a constructive proof of the theorem, which applies both to smooth and analytic hamiltonians and may be generalised to infinite-dimensional systems
Summary
In [Eli, Eli90] Eliasson suggested a constructive proof of the theorem, which applies both to smooth and analytic hamiltonians and may be generalised to infinite-dimensional systems. A hamiltonian of any equation from the KdV hierarchy is an analytic function of I (provided that m is so big that this hamiltonian is analytic on the space H0m); c) the maps Ψ, corresponding to different m, agree. It specifies the result by stating that a local transformation which integrates the KdV hierarchy may be chosen ‘1-smoother than its linear part’. This specification is crucial to study qualitative properties of perturbed KdV equations, e.g. see [KP09].
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