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Monotonicity properties of blow-up time for nonlinear Schr\"{o}dinger equation: numerical tests

Christophe Besse·Rémi Carles·Norbert Mauser·Hans-Peter Stimming
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Abstract

We consider the focusing nonlinear Schr\"{o}dinger 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.

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Christophe Besse, Rémi Carles, Norbert Mauser, Hans-Peter Stimming. 2006-03-04. Monotonicity properties of blow-up time for nonlinear Schr\"{o}dinger equation: numerical tests. https://arxiv.org/abs/math/0603107

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