Interaction of coherent states for Hartree equations

TL;DR

This paper studies how two initial wave packets evolve and interact under the Hartree equation with a smooth kernel and external potential in the semiclassical regime, revealing different nonlinear effects depending on the nonlinearity strength.

Contribution

It provides a detailed analysis of the interaction of coherent states in the Hartree equation, identifying three types of nonlinear phenomena and describing their effects on wave packet structure.

Findings

Wave packets maintain their coherent state structure during interaction.

Nonlinear effects influence the envelope and phase space centers of wave packets.

Different nonlinear regimes produce distinct interference phenomena.

Abstract

We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets, and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.