The Self-Focusing Singularity in the Nonlinear Schrödinger Equation

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like(log Δt)/Δt]^1/2, where Δt is the time remaining to the appearance of the singularity.