01347nas a2200289 4500008004100000022001300041245008600054210006900140300001200209490000700221520039800228653002600626653002200652653002800674653002500702653001700727653002500744653002200769653002100791653002800812653001900840653001900859100002400878700001700902700002100919856011700940 2013 eng d a1120633000aExistence and stability of quasi-periodic solutions for derivative wave equations0 aExistence and stability of quasiperiodic solutions for derivativ a199-2140 v243 aIn this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.10aConstant coefficients10aDynamical systems10aExistence and stability10aInfinite dimensional10aKAM for PDEs10aLinearized equations10aLyapunov exponent10aLyapunov methods10aQuasi-periodic solution10aSmall divisors10aWave equations1 aBerti, Massimiliano1 aBiasco, Luca1 aProcesi, Michela uhttps://www.math.sissa.it/publication/existence-and-stability-quasi-periodic-solutions-derivative-wave-equations01376nas 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-kdv