January  2008, 9(1): 11-36. doi: 10.3934/dcdsb.2008.9.11

Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence

1. 

Laboratoire Paul Painlevé, UMR CNRS 8524, Université de Sciences et Technologies de Lille, Cité Scientifique, F-59655 Villeneuve d'Ascq Cedex, France

2. 

Institut CNRS Pauli, UMI 2842 & Université Montpellier 2, Mathématiques, CC 051, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France

3. 

Wolfgang Pauli Institute c/o Fak. f. Mathematik, Univ. Wien, Nordbergstr. 15, A-1090 Wien, Austria, Austria

Received  October 2006 Revised  June 2007 Published  October 2007

We consider focusing nonlinear Schrödinger equations (NLS), in the $L^2$-critical and supercritical cases. We present a systematic numerical investigation of the dependence of the blow-up time on properties of the data or on the (parameters of the) equation in three cases: dependence on the strength of the nonlinearity in the equation when the initial data is fixed; dependence on the strength of a damping term in the equation when the initial data is fixed; and dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed. For some cases, analytic results are available and presented. In most situations our numerical counterexamples show that monotonicity in the evolution of the blow-up time does not occur. In addition they show that in certain regimes the blow-up time is very sensitive to the different parameters that we modulate.
    Our numerical solutions are very reliable since not only we test independence on the precise setting of the numerical problem (size of the periodic domain, discretization etc.) but we compare the same simulations with two different methods in two independent codes: a spectral time splitting code and a relaxation method, with results identical at the order of precision.
Citation: Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
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