Coherent states for systems of $L^2-$supercritical nonlinear Schr\"odinger equations

TL;DR

This paper studies the evolution of wave packets in matrix-valued nonlinear Schr"odinger equations, demonstrating their persistence in eigenspaces and establishing a superposition principle in the semi-classical limit.

Contribution

It introduces a framework for analyzing coherent states in matrix-valued nonlinear Schr"odinger equations, including propagation within eigenspaces and a superposition principle.

Findings

Wave packets remain in the same eigenspace during evolution.

A nonlinear superposition principle is established.

Results hold under long-range assumptions for the potential.

Abstract

We consider the propagation of wave packets for a nonlinear Schr\"odinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.