Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrodinger equation

We consider the following wave guide nonlinear Schrodinger equation, on the spatial cylinder . We establish a modified scattering theory between small solutions to this equation and small solutions to the cubic SzegA equation. The proof is an adaptation of the method of Hani et al. (Modified scattering for the cubic Schrodinger equation on product spaces and applications, 2015.). Combining this scattering theory with a recent result by G,rard and Grellier (On the growth of Sobolev norms for the cubic SzegA equation, 2015), we infer existence of global solutions to (WS) which are unbounded in the space for every .