Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

• 成果类型: Article
• 署名作者: Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; University of Toronto; Massachusetts Institute of Technology (MIT); Kobe University; University of California System; University of California Los Angeles
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0242-2
• 发表日期: 2010
• 页码: 39-113
• 关键词: birkhoff normal-form; quasi-periodic solutions; arnold diffusion; sobolev norms; hamiltonian perturbations; weak turbulence; GROWTH; systems; oscillations; STABILITY

摘要

We consider the cubic defocusing nonlinear Schrodinger 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 delta << 1, K >> 1, s > 1, we construct smooth initial data u(0) with parallel to u(0)parallel to(s)(H) < delta, so that the corresponding time evolution u satisfies parallel to u(T)parallel to(Hs) > K at some time T. This growth occurs despite the Hamiltonian's bound on parallel to u(t)parallel to((H) over dot1) and despite the conservation of the quantity parallel to u(t)parallel to(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.