# Unbounded orbits for the cubic nonlinear Schrodinger equation in the semi periodic setting

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A natural question in the study of nonlinear dispersive equations is to describe their asymptotic behavior. In the Euclidean plane, in great generality, global solutions scatter (i.e. asymptotically follow a linear flow). In a bounded domain, the energy cannot escape to infinity and one expects that nonlinear effects prevent the solutions from settling'' to some nice simple dynamics. But really no one knows for sure.  It has been proposed that typical solutions visit all of the phase space, (except for the trivial limitations provided by a few conservation laws). A weaker statement is the question of the existence of one solution whose Sobolev norms H^s for s>2 grows unboundedly. This is still open for the Torus T^2 despite exciting developments by Colliander-Keel-Staffilani-Takaoka-Tao and Guardia-Kaloshin. However, it can be realized if one looks only at partially periodic solutions in 3 dimensions.