New self-similar solutions of the nonlinear Schrodinger equation with moving mesh computations

We study the blow-up self-similar solutions of the radially symmetric nonli near Schrodinger equation (NLS) given by iu(t) + u(rr) + d - 1/ru(r) + 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 anal ytic results, we demonstrate that there is a countably infinite set of blow -up self-similar solutions which satisfy a second order complex ordinary di fferential equation with an integral constraint. These solutions are charac terised 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 solution s is then studied by solving the NLS by using an adaptive numerical method. This method uses moving mesh partial differential equations and exploits t he scaling invariance properties of the underlying equation. We demonstrate that the single-humped selfsimilar solution is globally stable whereas the multi-humped solutions all appear to be unstable. (C) 1999 Academic Press.