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 $Δ_0$ 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 $Δ_0$ is less than a critical value $Δ_c$, then the packet will propagate in time with linearly growing width but for $Δ>Δ_c$, the packet will start becoming narrow and cease to be a Gaussian . For a simple harmonic oscillator, we find that for $Δ_0$ 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.

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