Massimiliano Berti, Thomas Kappeler, Riccardo Montalto
Large KAM tori for quasi-linear perturbations of KdV
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ABSTRACT.  We prove the persistence of most finite gap solutions of the KdV equation on the circle under sufficiently small and smooth quasi-linear Hamiltonian perturbations. The proof makes use of suitable symplectic coordinates, introduced by Kappeler-Montalto [15], in the vicinity of any finite-gap manifold, which admit a pseudo-differential expansion. Then we implement a Nash-Moser iteration scheme. A key step is to diagonalize the linearized operators, obtained at any approximate quasi-periodic solution, with sharp asymptotic estimates of their eigenvalues.