New Self-Similar Solutions of the Nonlinear Schrödinger Equation with Moving Mesh Computations

We study the blow-up self-similar solutions of the radially symmetric nonlinear Schrodinger equation (NLS) given by iut+urr+d?1rur+u|u|2, with dimension d>2. These solutions become infinite in a finite time T. By a series of careful numerical computations, partly supported by analytic results, we demonstrate that there is a countably infinite set of blow-up self-similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint. These solutions are characterised by the number of oscillations in their amplitude when d is close to 2. The solutions are computed as functions of d and their behaviour in the critical limit as d?2 is investigated. The stability of these solutions is then studied by solving the NLS by using an adaptive numerical method. This method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation. We demonstrate that the single-humped self-similar solution is globally stable whereas the multi-humped solutions all appear to be unstable.