Energy cascades for NLS on $\T^d$

Rémi Carles (1) , Erwan Faou (2, 3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IPSO - Invariant Preserving SOlvers

We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.

Domaines: Equations aux dérivées partielles [math.AP]

Rémi Carles, Erwan Faou. Energy cascades for NLS on $\T^d$. 2010. ⟨hal-00528792v1⟩