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

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

1 SIMPAF - SImulations and Modeling for PArticles and Fluids
2 MAB - Laboratoire de Mathématiques Appliquées de Bordeaux
3 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. Discrete and Continuous Dynamical Systems - Series B, 2008, 9 (1), pp.11-36. ⟨10.3934/dcdsb.2008.9.11⟩. ⟨hal-00020062v2⟩