Potential scattering problems governed by the time-dependent Gross-Pitaevskii equation are investigated numerically for various values of coupling constants. The initial condition is assumed to have the Gaussian-type envelope, which differs from the soliton solution. The potential is chosen to be a box or well type. We estimate the dependences of reflectance and transmittance on the width of the potential and compare these results with those given by the stationary Schrödinger equation. We attribute the behaviors of these quantities to the limitation on the width of the nonlinear wave packet.The coupling constant and the width of the potential play an important role in the distribution of the waves appearing in the final state of scattering.

The nonlinear Schrödinger (NLS) equation under the box-type initial condition, which models general multiple pulses deviating from pure solitons, is analyzed. Following the approximation by splitting the initial pulse into many small bins [G. Boffetta and A. R. Osborne, J. Comp. Phys. 102, 25 (1992)], we can analyze the Zakharov-Shabat eigenvalue problem to construct transfer matrices connecting the Jost functions in each interval without direct numerical computation. We can obtain analytical expressions for the scattering data that describe interfering radiation emitted from initial pulses. The number of solitons that appear in the final stage is predicted theoretically, and the condition generating an unusual wave such as a double-pole soliton is derived. Numerical analyses under box-type initial conditions are also performed to show that the interplay between the tails from decaying pulses can affect the asymptotic profile.

We propose an indirect method for observing radiation from an incomplete soliton with a sufficiently large amplitude. We show that the radiation causes a notched structure on the envelope of the wave packet in the momentum space. The origin of this structure is the interference between the main body of oscillating solitons and the small radiation in the momentum space. We numerically integrate the nonlinear Schrödinger equation and perform Fourier transformation to confirm that the predicted structure really appears. We also show a simple model which reproduces the qualitative result. The experimental detection of the notched structure with the Bose-Einstein condensation of neutral atoms is discussed and suitable parameters for this detection experiment are shown.

We present a novel approximation method that can predict the number of solitons asymptotically appearing under arbitrary rapidly decreasing initial wave packets. The number of solitons can be estimated without integration of the original soliton equations. As an example, we take the one-dimensional nonlinear Schrödinger equation and estimate the behaviors of the scattering amplitude in detail. The results show good agreement compared with those obtained by direct numerical integration. The presented method is applicable to a wide class of one-dimensional soliton equations.