## Abstract

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1, K≫1, s<1, we construct smooth initial data u_{0} with {double pipe} u_{0} {double pipe} H ^{s} < δ, so that the corresponding time evolution u satisfies {double pipe} U(T) {double pipe} H ^{s} > K at some time T. This growth occurs despite the Hamiltonian's bound on {double pipe} u(t) {double pipe} Ḣ ^{1} and despite the conservation of the quantity {double pipe} u(t) {double pipe} 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.

Original language | English (US) |
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Pages (from-to) | 39-113 |

Number of pages | 75 |

Journal | Inventiones Mathematicae |

Volume | 181 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2010 |

### Bibliographical note

Funding Information:J.C. is supported in part by N.S.E.R.C. grant RGPIN 250233-07.

Funding Information:

M.K. is supported in part by the Sloan Foundation and N.S.F. Grant DMS0602792.

Funding Information:

T.T. is supported by NSF grant DMS-0649473 and a grant from the Macarthur Foundation.