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 \(\|u_{0}\|_{{H}^{s}}<\delta\), 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.
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J.C. is supported in part by N.S.E.R.C. grant RGPIN 250233-07.
M.K. is supported in part by the Sloan Foundation and N.S.F. Grant DMS0602792.
G.S. is supported in part by N.S.F. Grant DMS 0602678.
H.T. is supported in part by J.S.P.S. Grant No. 13740087.
T.T. is supported by NSF grant DMS-0649473 and a grant from the Macarthur Foundation.
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Colliander, J., Keel, M., Staffilani, G. et al. Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. math. 181, 39–113 (2010). https://doi.org/10.1007/s00222-010-0242-2
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DOI: https://doi.org/10.1007/s00222-010-0242-2