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

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 < δ, 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) Ḣ 1 and despite the conservation of the quantity u(t) L 2 .

Globally Regular Instability of 3-Dimensional Anti–De Sitter Spacetime

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Three-dimensional Gravity and Instability of AdS$_{3}$

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Non-Scattering of the Solution of the Nonlinear Schr&#246;dinger Equation on the Torus

“…which approach a solution to the linear equation at time t = ∞. Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T. [5] consider the cubic nonlinear Schrödinger equation on two dimensional torus, and prove the solution cannot scatter to free solution in 1 ( ).…”