Beating effects in cubic Schrödinger systems and growth of Sobolev norms

Benoît Grébert (1) , Eric Paturel (1) , Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

We consider a system of coupled cubic Schrödinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schrödinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times.

Mots clés: Norm inflation; Nonlinear Schrödinger system; Resonant normal form; energy exchange; Linear Schrödinger equation; Time-dependent potential; Norm inflation.

Domaines: Equations aux dérivées partielles [math.AP]

Benoît Grébert, Eric Paturel, Laurent Thomann. Beating effects in cubic Schrödinger systems and growth of Sobolev norms. Nonlinearity, 2013, 26 (5), pp.1361--1376. ⟨hal-00726057⟩