An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifold

2013/11/27 by Massimiliano Berti, Livia Corsi, Michela Procesi · 1 citation
Mathematics · #math.AP #math.FA #msc:35L05 #msc:35Q55 #msc:37K55 #msc:58C15

paper · pdf · doi:10.1007/s00220-014-2128-4

45 pages

arxiv created 2013/11/27 · arxiv updated 2015/06/18

Abstract

We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

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