Numerical Calculation of Monotonicity Properties of the Blow-Up Time of NLS

We investigate blow-up of the focusing nonlinear Schr ¨odinger equation, in the critical and supercritical cases. Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on properties of the data or the equation. Three cases are considered: dependence on the scale of the nonlinearity 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; and dependence upon a damping factor when the initial data are fixed. In most of these situations, monotonicity in the evolution of the blow-up time does not occur.