This research focuses on the scattering of particles on a nonlinear potential well in both conservative and dissipative regimes. In a one-dimensional conservative case involving two coupled particles, scattering can lead to two distinct outcomes: transition and reflection, with the outcome itself demonstrating a strong correlation with energy transfer processes within the system. In general, the system exhibits strong sensitivity to initial conditions, causing the outcomes to vary wildly with small alterations in system parameters. Under asymptotic conditions, the system yields robust and predictable results, particularly for small coupling-free length or large initial velocities, where analytical approaches accurately predict energy transfer trends within the system.
In the two-dimensional case, with a single particle, the scattering outcomes become non-binary, and the particle may end up in an infinite number of possible trajectories outside the well. A method for calculating the scattering cross-section for potential wells (and hills) is presented, along with several potential forms for which analytical solutions can be derived.
