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

By James E. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao. 2010

302 citations recorded by the source corpus.

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 $\|u_{0}\|_{{H}^{s}}<\delta$, 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5f7511aee227ea16b177696569c240f11706ed1b

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Citations in the corpus

An explicit formula for the cubic Szegö equation (2013, unknown): “This induces the following instability of H s norms , a phenomenon of the same nature as the one displayed by Colliander, Keel, Staﬃlani, Takaoka and Tao in [1].”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “13Also notice that the energy curve h = 0 supports only two types of orbit namely that given by cos 2φ = − 1 2 and ṙ = ±2 √ 3r(1− r) which leads to the heteroclinic orbit at the basis of the construction in [31].”
Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations (2012, methodology): “While the proof of this theorem relies heavily on the constructions in [11], it offers a stark example of some of the difficulties encountered as one tries to prove long-time strong instability for (2.”
The cubic Szegő equation and Hankel operators (2015, background): “row at most polynomially in time. Existence of unbounded trajectories only recently [24] received a positive answer in some very special cases, while several instability results were already obtained [9], [22], [23], [26], [21]. Natural model problems for CHAPTER 1.INTRODUCTION 2 studying these phen”
A KAM algorithm for the resonant non-linear Schrödinger equation (2012, background): “Unfortunately this very strong property holds only for the cubic NLS in dimension one, indeed for d > 1 the non-integrability of the NLS normal form has been exploited (see for instance [6] and [14]) to construct diffusive orbits.”
A Normal Form for the Schrödinger Equation with Analytic Non-linearities (2010, background): “Suitable non-generic choices of S lead also to interesting non-integrable behavior as for instance in the paper [7].”