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