Monotonicity properties of blow-up time for nonlinear Schrödinger equation: numerical tests

Christophe Besse (1, 2) , Rémi Carles (3) , Norbert Mauser (4) , Hans-Peter Stimming (4)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 SIMPAF - SImulations and Modeling for PArticles and Fluids
3 MAB - Laboratoire de Mathématiques Appliquées de Bordeaux
4 ICP - Institut CNRS-PAULI

We consider the focusing nonlinear Schrödinger equation, in the $L^2$-critical and supercritical cases. We investigate numerically the dependence of the blow-up time on a parameter in three cases: dependence upon the coupling constant, when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; finally, dependence upon a damping factor when the initial data are fixed. It turns out that in most situations monotonicity in the evolution of the blow-up time does not occur. In the case of quadratic oscillations in the initial data, with critical nonlinearity, monotonicity holds; this is proven analytically.

Mots clés: Blow-up time; nonlinear Schrödinger equation; numerical experiments

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Analyse numérique [math.NA]

Christophe Besse, Rémi Carles, Norbert Mauser, Hans-Peter Stimming. Monotonicity properties of blow-up time for nonlinear Schrödinger equation: numerical tests. 2006. ⟨hal-00020062v1⟩