Nonlinear coherent states and Ehrenfest time for Schrodinger equation

TL;DR

This paper investigates the propagation of wave packets in the nonlinear Schrödinger equation within the semi-classical limit, identifying a critical initial size that determines the influence of nonlinearity up to the Ehrenfest time.

Contribution

It establishes the existence of a critical initial data size in relation to the Planck constant and describes the wave function's behavior as a nonlinear coherent state up to Ehrenfest time.

Findings

Wave packets propagate as nonlinear coherent states under certain initial conditions.

A critical initial size determines when nonlinearity affects the wave evolution.

A nonlinear superposition principle for wave packets is proven.

Abstract

We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.