TY - JOUR

T1 - Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

AU - Colliander, J.

AU - Keel, M.

AU - Staffilani, G.

AU - Takaoka, H.

AU - Tao, T.

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010/7

Y1 - 2010/7

N2 - 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 u0 with {double pipe} u0 {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} L2. 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.

AB - 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 u0 with {double pipe} u0 {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} L2. 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.

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U2 - 10.1007/s00222-010-0242-2

DO - 10.1007/s00222-010-0242-2

M3 - Article

AN - SCOPUS:77952799350

VL - 181

SP - 39

EP - 113

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -