Self-Similarity Analysis of the Nonlinear Schrödinger Equation in the Madelung Form

Figure 1

Various shape functions of the density. The blue curve is the analytic solution of equation (7) with nonlinearity parameter. Black, red, and green functions are numerical results of equation (10) for and nonlinearity parameters. (All curves are for the same initial conditions of and .) Note that larger parameters mean larger deviation from the analytic result.