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Quasi-periodic solutions for quasi-linear generalized KdV equations

TitleQuasi-periodic solutions for quasi-linear generalized KdV equations
Publication TypeJournal Article
Year of Publication2017
AuthorsGiuliani, F
JournalJournal of Differential Equations
Pagination5052 - 5132
KeywordsKAM for PDE's; KdV; Nash–Moser theory; Quasi-linear PDE's; Quasi-periodic solutions

We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash–Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.


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