We prove the existence of Cantor families of small amplitude, linearly
stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian
generalized KdV equations. We consider the most general quasi-linear quadratic
nonlinearity. The proof is based on an iterative Nash-Moser algorithm. To
initialize this scheme, we need to perform a bifurcation analysis taking into
account the strongly perturbative effects of the nonlinearity near the origin.
In particular, we implement a weak version of the Birkhoff normal form method.
The inversion of the linearized operators at each step of the iteration is
achieved by pseudo-differential techniques, linear Birkhoff normal form
algorithms and a linear KAM reducibility scheme.Comment: arXiv admin note: substantial text overlap with arXiv:1404.3125,
arXiv:1508.02007, arXiv:1602.02411 by other author
