  We consider the cubic defocusing nonlinear Schr\"odinger equation on the two
dimensional torus. We exhibit smooth solutions for which the support of the
conserved energy moves to higher Fourier modes. This weakly turbulent behavior
is quantified by the growth of higher Sobolev norms: given any delta << 1, K
>>1, s >1, we construct smooth initial data u_0 with ||u_0||_{{H}^s} < delta,
so that the corresponding time evolution u satisfies ||u(T)||_{{H}^s} > K at
some time T. This growth occurs despite the Hamiltonian's bound on
||u(t)||_{\dot{H}^1} and despite the conservation of the quantity
||u(t)||_{L^2}.
  The proof contains two arguments which may be of interest beyond the
particular result described above. The first is a construction of the
solution's frequency support that simplifies the system of ODE's describing
each Fourier mode's evolution. The second is a construction of solutions to
these simpler systems of ODE's which begin near one invariant manifold and
ricochet from arbitrarily small neighborhoods of an arbitrarily large number of
other invariant manifolds. The techniques used here are related to but are
distinct from those traditionally used to prove Arnold Diffusion in
perturbations of Hamiltonian systems.
