Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation

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dc.creatorColliander, J.
dc.creatorKeel, M.
dc.creatorStaffilani, G.
dc.creatorTakaoka, H.
dc.creatorTao, T.
dc.date2008-08-13
dc.date2008-08-18
dc.date.accessioned2026-07-07T09:56:51Z
dc.date.available2026-07-07T09:56:51Z
dc.descriptionWe 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 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.
dc.description54 pages, 7 figures. 5 typographical errors corrected
dc.identifierhttps://arxiv.org/abs/0808.1742
dc.identifierhttp://arxiv.org/abs/0808.1742
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/167127
dc.subjectAnalysis of PDEs
dc.subjectDynamical Systems
dc.titleWeakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation
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