Dario Bambusi (Università degli studi di Milano)
It is well known that solutions of the Nonlinear Klein Gordon equation $$\frac{1}{c^2}u_{tt}-u_{xx}+c^2 u\pm u^3=0\ ,\quad u(0,t)=u(\pi,t)=0$$ are well approximated in the non relatistic limit ($c\to\infty$)by solutions of the cubic Nonlinear Schr\”odinger equation.
In the present talk I will present a result based on KAM theory, according to which the quasiperiodic solutions of NLKG constructed through KAM theory converge {\it uniformly for $t\in\R$} to solutions of NLS.
I will start the presentation by recalling some classical results on the justification of the NLS as the classical limit of NLKG, then I will give a precise statement of the result and the main ideas of the proof.
Joint work with Andrea Belloni and Filippo Giuliani.