BIRKHOFF COORDINATES OF INTEGRABLE HAMILTONIAN SYSTEMS IN ASYMPTOTIC REGIMES

In this thesis we investigate two examples of infinite dimensional integrable Hamiltonian systems in $1$-space dimension: the Toda chain with periodic boundary conditions and large number of particles, and the Korteweg-de Vries (KdV) equation on $\R$.

In the first part of the thesis we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number $N$ of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius $R/N^\alpha$
(in discrete Sobolev-analytic norms) into a ball of radius $R'/N^\alpha$ (with $R,R'>0$ independent of $N$) if and only if
$\alpha\geq2$. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size $R/N^2$, $0
In the second part of the thesis we study the scattering map of the KdV on $\R$. We prove that in appropriate weighted Sobolev spaces of the form $H^{N} \cap L^2_M$, with integers $N \geq 2M \geq 8$ and in the case of no bound states, the scattering map is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.