KAM for autonomous quasi-linear perturbations of KdV

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian differentiable perturbations of KdV. This is the first result that extends KAM theory to quasi-linear autonomous and parameter independent PDEs. The core of the proof is to find an approximate inverse of the linearized operators at each approximate solution and to prove that it satisfies tame estimates in Sobolev spaces. A symplectic decoupling procedure reduces the problem to the one of inverting the linearized operator restricted to the normal directions. For this aim we use pseudo-differential operator techniques to transform such linear PDE into an equation with constant coefficients up to smoothing remainders. Then a linear KAM reducibility technique completely diagonalizes such operator. We introduce the “initial conditions” as parameters by performing a “weak” Birkhoff normal form analysis, which is well adapted for quasi-linear perturbations.