Giuliani, Filippo2016-07-122016-07-122016-07https://openscience.sissa.it/handle/1963/35204We 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.enQuasi-linear Partial differential equationsQuasi-periodic solutionsNash-Moser theoryKAM for PDE'sPreprintMAT/0538/2016/MATE