Wave packet dynamics for a non-linear Schrodinger equation: Qualitative changes with changes in the initial width

The propagation of an initially Gaussian wave packet of width ${\mathrm{\Delta }}_0$ in a cubic non-linear Schrodinger equation with a negative coupling constant for the nonlinear term is considered . It is predicted analytically and verified numerically that for a free particle if ${\mathrm{\Delta }}_0$ is less than a critical value ${\mathrm{\Delta }}_c$, then the packet will propagate in time with linearly growing width but for $\mathrm{\Delta }>{\mathrm{\Delta }}_c$, the packet will start becoming narrow and cease to be a Gaussian . For a simple harmonic oscillator, we find that for ${\mathrm{\Delta }}_0$ smaller than a critical value, there always exist a coupling strength for which the packet simply oscillates about the mean position without changing its shape.