A note on KAM theory for quasi-linear and fully nonlinear forced KdV

We present recent results concerning quasi-periodic solutions for quasilinear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of different ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a differential operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coefficients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.