01519nas a2200145   4500008004100000022001300041245008300054210006900137300000900206520098400215100001301199700002401212700002301236856011401259       2014                        eng d  a0025583100aKAM for quasi-linear and fully nonlinear forced perturbations of Airy equation0 aKAM for quasilinear and fully nonlinear forced perturbations of   a1-663 aWe prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an 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. At this point we implement a Nash-Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients. © 2014 Springer-Verlag Berlin Heidelberg.1 aBaldi, P1 aBerti, Massimiliano1 aMontalto, Riccardo  uhttps://www.math.sissa.it/publication/kam-quasi-linear-and-fully-nonlinear-forced-perturbations-airy-equation