12-147 Pietro Baldi, Massimiliano Berti, Riccardo Montalto

KAM for quasi-linear and fully nonlinear forced KdV (994K, PDF) 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.

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