Wave packet dynamics for Gross–Pitaevskii equation in one dimension: Dependence on initial conditions

Abstract

The propagation of an initially Gaussian wave packet of width Δ0 in Gross–Pitaevskii equation is extensively studied both for attractive and repulsive interactions. It is predicted analytically and verified numerically that for a free particle with attractive interaction, the dynamics of the width is governed by an effective potential which is sensitive to initial conditions. If Δ0 is equal to a corresponding critical width Δc, then the packet will propagate in time with very little change in shape. These are in essence like coherent states. Whereas, if Δ≠Δc, depending on the nature of the effective potential for chosen Δ0 and the interaction strength (|g|), the width of the packet in course of time, either oscillates with bounded width or will spread like a free particle. For a simple harmonic oscillator (SHO) also, we find that for Δ0 smaller than a critical value, there always exists a coupling strength for which the packet simply oscillates about the mean position without changing its shape, once again providing a resemblance to a coherent state. We also consider the Morse potential, which interpolates between the free particle and the oscillator. For large attractive interactions, the two limiting dynamics (free and simple harmonic) are indeed observed but in the intermediate form of the potential where the nonlinear terms dominate in the dynamics, an initial Gaussian wave packet does not retain its shape. For repulsive interaction, the Gaussian packet always changes shape no matter what the system parameters are.