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We consider the cubic defocusing nonlinear Schr\"odinger equation on the twodimensional torus. We exhibit smooth solutions for which the support of theconserved energy moves to higher Fourier modes. This weakly turbulent behavioris 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 atsome 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 theparticular result described above. The first is a construction of thesolution's frequency support that simplifies the system of ODE's describingeach Fourier mode's evolution. The second is a construction of solutions tothese simpler systems of ODE's which begin near one invariant manifold andricochet from arbitrarily small neighborhoods of an arbitrarily large number ofother invariant manifolds. The techniques used here are related to but aredistinct from those traditionally used to prove Arnold Diffusion inperturbations of Hamiltonian systems.

Weakly turbulent solutions for the cubic defocusing nonlinear Schr\"odinger equation