- 12-147 Pietro Baldi, Massimiliano Berti, Riccardo Montalto
- KAM for quasi-linear and fully nonlinear forced KdV
Nov 28, 12
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Abstract. We prove the existence of quasi-periodic, small amplitude, solutions for quasi-linear and fully
nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain
the linear stability of the solutions. The proofs are based on a combination of di erent ideas and
techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates
the linearized operator to a di fferential operator with constant coeffcients plus a bounded remainder.
These transformations are obtained by changes of variables induced by
diffeomorphisms of the torus and
pseudo-di fferential 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.