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 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes

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 $\T^d$. Discrete and Continuous Dynamical Systems - Series S, 2012, 32 (6), pp.2063-2077. ⟨10.3934/dcds.2012.32.2063⟩. ⟨hal-00528792v2⟩