Mathematics

I will present a result on the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. I considered the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash-Moser algorithm. To initialize this scheme, one needs to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, I implemented 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.