Roberto Feola's seminar at SISSA

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENT


*Title:*/ Quasi-periodic solutions for fully nonlinear NLS/

Roberto Feola, SISSA,

Tuesday, 24 November, room 133,  14-15

*Abstract: *We consider a class of fully nonlinear, autonomous and 
reversible Schrödinger equations on the circle and  we prove the 
existence and the stability of Cantor families of (small amplitude) 
quasi-periodic solutions. The proof is based on a combination of 
different ideas:  (i) we perform a “weak” Birkhoff normal form step in 
order to find an approximately invariant manifold on which the dynamics 
is approximately integrable; (ii) we introduce a suitable generalization 
of a KAM nonlinear iteration for “tame” and ''unbounded'' vector fields 
  based on the invertibility of the linearized equation in a 
neighborhood of the origin; (iii) we exploit the  “pseudo-differential 
structure” of the vector field and we prove the invertibility of the 
linearized operator, using a regularization procedure which conjugates 
the operator to a differential operator with constant coefficients plus 
a bounded remainder. This latter step is obtained through 
transformations generated by torus diffeomorphisms and 
pseudo-differential operators. Then we use a KAM-like reducibility 
scheme that reduces to constant coefficients the linearized operator at 
the solution. This gives the linear stability.