01376nas a2200145 4500008004100000245007300041210006900114260003400183520083400217653001701051100001301068700002401081700002301105856010201128 2013 en d00aA note on KAM theory for quasi-linear and fully nonlinear forced KdV0 anote on KAM theory for quasilinear and fully nonlinear forced Kd bEuropean Mathematical Society3 aWe present the recent results in [3] concerning quasi-periodic solutions
for quasi-linear and fully nonlinear forced perturbations of KdV equations.
For Hamiltonian or reversible nonlinearities the solutions are linearly stable.
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 erential operator
with constant coe cients plus a bounded remainder. These transformations
are obtained by changes of variables induced by di eomorphisms of the torus
and pseudo-di erential operators. (iii) A reducibility KAM scheme, which
completes the reduction to constant coe cients of the linearized operator,
providing a sharp asymptotic expansion of the perturbed eigenvalues.10aKAM for PDEs1 aBaldi, P1 aBerti, Massimiliano1 aMontalto, Riccardo uhttps://www.math.sissa.it/publication/note-kam-theory-quasi-linear-and-fully-nonlinear-forced-kdv01454nas a2200145 4500008004100000022001400041245009100055210007000146260000900216490000800225520090000233100002401133700002101157856013001178 2011 eng d a0012-709400aNonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces0 aNonlinear wave and Schrödinger equations on compact Lie groups a c20110 v1593 aWe develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.1 aBerti, Massimiliano1 aProcesi, Michela uhttps://www.math.sissa.it/publication/nonlinear-wave-and-schr%C3%B6dinger-equations-compact-lie-groups-and-homogeneous-spaces00389nas a2200109 4500008004100000245007400041210006900115260001300184100002400197700002200221856003600243 2001 en d00aNon-compactness and multiplicity results for the Yamabe problem on Sn0 aNoncompactness and multiplicity results for the Yamabe problem o bElsevier1 aBerti, Massimiliano1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/1345