Abstract
We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear (also called strongly nonlinear) autonomous Hamiltonian differentiable perturbations of the mKdV equation. The proof is based on a weak version of the Birkhoff normal form algorithm and a nonlinear Nash–Moser iteration. The analysis of the linearized operators at each step of the iteration is achieved by pseudo-differential operator techniques and a linear KAM reducibility scheme.
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Acknowledgments
This research was supported by the European Research Council under FP7, ERC Project 306414 HamPDEs, and PRIN 2012 “Variational and perturbative aspects of nonlinear differential problems”. This research was carried out in the frame of Programme STAR, financially supported by UniNA and Compagnia di San Paolo.
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Baldi, P., Berti, M. & Montalto, R. KAM for autonomous quasi-linear perturbations of mKdV. Boll Unione Mat Ital 9, 143–188 (2016). https://doi.org/10.1007/s40574-016-0065-1
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DOI: https://doi.org/10.1007/s40574-016-0065-1