Computation of Self-similar Solution Profiles for the Nonlinear Schrödinger Equation

We discuss the numerical computation of self-similar blow-up solutions of the classical nonlinear Schrödinger equation in three space dimensions. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. The problem of the computation of this self-similar solution profile reduces to a nonlinear, ordinary differential equation on an unbounded domain. We show that a transformation of the independent variable to the interval [0,1] yields a well-posed boundary value problem with an essential singularity. This can be stably solved by polynomial collocation. Moreover, a Matlab solver developed by two of the authors can be applied to solve the problem efficiently and provides a reliable estimate of the global error of the collocation solution. This is possible because the boundary conditions for the transformed problem serve to eliminate undesired, rapidly oscillating solution modes and essentially reduce the problem of the computation of the physical solution of the problem to a boundary value problem with a singularity of the first kind. Furthermore, this last observation implies that our proposed solution approach is theoretically justified for the present problem.